Bounds on option prices for semimartingale market models^{∗}

Alexander A. Gushchin^{†}

Ernesto Mordecki^{‡}

December 27, 2001

Abstract

We propose a methodology for the determination of the range of option prices of a European option in a general semimartingale market model, with a convex payoff function. Prices are obtained as expectations along the set of equivalent martingale measures.

Since the set of prices is an interval on the real line, two main questions are considered: (i) how to find upper and lower estimates for the range of prices, and (ii) how to establish the attainability of these estimates. To solve the first question, we introduce a partial ordering in the set of distributions of the discounted stock prices (adapted from the theory of statistical experiments), which allows us to find extremal distributions and, correspondingly, upper and lower bounds for the range of option prices. Weak convergence of probability measures is used to answer the second question, whether the bounds obtained at the first step are exact.

Exploiting stochastic calculus, we give answers to both questions in (the most natural for this problem) terms of predictable characteristics of the stochastic logarithm of the discounted stock price process. Particular attention is given to two examples: discrete time and diffusion with jumps market models.

∗The first author’s research was partially supported by INTAS Grant 97-30204.

†Steklov Mathematical Institute, Moscow, Russia ‡Facultad de Ciencias, Centro de Matemática, Montevideo, Uruguay

1 Introduction

1.1. Consider a mathematical model of a financial market with two assets. The first one is a non-risky asset B = {B_{t}}_{0≤t≤T} with a deterministic interest rate r = r(t), such that

B_{t} = B_{0}e^{∫_{0}^{t} r(s)ds}, 0 ≤ t ≤ T.

The evolution of the second asset is modeled through a strictly positive semimartingale

S = {S_{t}}_{0≤t≤T} with S_{0} a positive constant,  (1.1)

defined on a stochastic basis B = (Ω, F, {F_{t}}_{0≤t≤T} , P). For simplicity we assume F = F_{T}. In this model, we consider a European option with maturity T and pay-off g(S_{T}), where

g: (0, ∞) → ℝ is a convex function.  (1.2)

Classical examples are European call and put options with payoff functions g(x) = (x − K)^{+} and g(x) = (K − x)^{+} respectively. Denote by P the set of equivalent martingale measures, that is

P = \{Q: Q probability measure, Q ∼ P, \{\frac{S_{t}}{B_{t}} \} is a Q-martingale\}.

We assume the absence of arbitrage in the sense P is not empty. It is well known, that if there exists only one martingale measure Q^{∗}, then the market is complete, and the price of the European option introduced above is given by E_{Q^{∗}}B_{T}^{−1}g(S_{T}), due to the possibility of perfect replication of the contingent claim g(S_{T}). In the case of an incomplete market, the situation we want to consider, the set P contains more than one measure. Then, in our view, the problem of pricing options raises two main questions. First, how to choose “good” martingale measures that give satisfactory pricing strategies, and second, a question of more theoretical character, which prices are possible when the set of all martingale measures is considered. In this paper we are concerned with the second question, the determination of the set of possible values for the option, i.e. admissible prices. The first remark is that, as P is a convex set, and the expectation is a linear functional, the set of admissible prices is convex in the real line, in other words, an interval. Based on this remark denoting γ(Q) = E_{Q}B_{T}^{−1}g(S_{T}), we are interested in the computation of the quantities,

ℂ^{∗} = sup{γ(Q): Q ∈ M}

and

ℂ_{∗} = inf{γ(Q): Q ∈ M},

where M is a subset of P, possibly M = P. It is important to remark that ℂ^{∗} is related to the super-replication price of the contingent claim, that is the smallest amount necessary to hold a self-financing strategy whose value at time T is not smaller than g(S_{T}). Indeed, by the optional decomposition theorem due to Kramkov (1996), see also El Karoui and Quenez (1995) or Föllmer and Kabanov (1998), the super-replication price is equal to sup γ(Q), where the supremum is taken over the set P_{loc} of all equivalent local martingale measures Q. It is easy to see that, if P ≠ ∅ and g is nonnegative, then sup{γ(Q): Q ∈ P} = sup{γ(Q): Q ∈ P_{loc}}, see e.g. Lemma 18 in Jakubenas (1998). The range of option prices for the simplest one-step model is considered e.g. in Shiryaev (1999), Chapter V, § 1c. A discrete (in time and space) market model with independent returns is studied in Melnikov (1999). Shataev (1998) considers a discrete-time model with conditionally Gaussian returns. See also a recent paper by Rüschendorf (2001) on the discrete-time case. For continuous stock prices related works are Avellaneda et al. (1995), El Karoui et al. (1998), Frey and Sin (1999). The main three references for our work are Bellamy and Jeanblanc (2000) where a mixed jump-diffusion model is considered, and Eberlein and Jacod (1997) and Jakubenas (1998) in the case where S is the exponential of a Lévy process. In this paper we accumulate certain ideas from above works and propose a general methodology for the problem under consideration.

1.2. Our approach is based on the following facts adapted from the theory of binary statistical experiments in a form convenient for our purposes. See Section 2 for the details, where some other results used in the paper are given. Here we introduce the notation which will be used throughout the paper. Let S be the set of all probability measures μ on (ℝ_{+}, B(R_{+})) (B(X) stands for the Borel σ-algebra on X) satisfying the inequality ∫ x μ(dx) ≤ 1 (the domain of integration ℝ_{+} = [0, ∞) is omitted for short). When speaking of topological properties of S, the weak topology on S is always considered, and the weak convergence is denoted by ⇒. By Fatou’s lemma, S is a compact set. Denote by C the class of all convex real-valued functions defined on (0, +∞). Given f ∈ C, define

f(0) = lim_{x↓0} f(x), \frac{f(∞)}{∞} = lim_{x→∞} \frac{f(x)}{x}.

Due to convexity of f both limits exist and belong to (−∞, +∞], and we can define for any μ ∈ S and f ∈ C the functional

J_{f}(μ) = ∫ f(x) μ(dx) + \frac{f(∞)}{∞}(1 − ∫ x μ(dx))  (1.3)

(with the convention 0 · ∞ = 0). The functional J_{f}(μ) is well defined and takes values in (−∞, +∞].

Proposition 1.1 The mapping S ∋ μ ⇝ J_{f}(μ) is lower semi-continuous. Moreover, it is continuous if

f(0) + \frac{f(∞)}{∞} < ∞.  (1.4)

Define a binary relation ≼ on the set S by μ′ ≼ μ if J_{f}(μ′) ≤ J_{f}(μ) for any f ∈ C. It can be shown that μ′ ≼ μ if and only if

∫ (a − x)^{+}μ′(dx) ≤ ∫ (a − x)^{+}μ(dx) for any a > 0.  (1.5)

Proposition 1.2 (a) The relation ≼ is a partial ordering on S.

(b) μ′ ≼ μ if and only if there exist random variables ξ and ξ′ on the same probability space with the distributions μ and μ′ respectively such that E(ξ | ξ′) ≤ ξ′ a.s. In particular, μ′ ≼ μ implies μ′({0}) ≤ μ({0}) and ∫ x μ′(dx) ≥ ∫ x μ(dx).

(c) The partially ordered set (S, ≼) is order complete, i.e. every nonempty subset T of S has the least upper bound sup T and the greatest lower bound inf T .

(d) sup S = δ_{{0}} and inf S = δ_{{1}}, where δ_{{x}} is the Dirac measure at x.

1.3. Let us return to the option pricing problem. We want to show that the preceding results suggest a certain way to find the upper and lower prices.

First, let us remark that considering a discounted stock price one can simplify notationally the problem. Namely, given S as in (1.1) and g as in (1.2), introduce a process Z = {Z_{t}}_{0≤t≤T} and a function f: (0, ∞) → ℝ by

Z_{t} := \frac{B_{0}S_{t}}{S_{0}B_{t}}  and f(x) := B_{T}^{−1}g(B_{0}^{−1}S_{0}B_{T} x).  (1.6)

Then f ∈ C, Z_{0} = 1, Z is a strictly positive Q-martingale if (and only if) Q ∈ P, and

γ(Q) = E_{Q}f(Z_{T}) = J_{f}(μ_{Q}), Q ∈ P,

where μ_{Q} := L(Z_{T} | Q) ∈ S is the distribution of Z_{T} under Q. Therefore, ℂ_{∗} and ℂ^{∗} are the infimum and the supremum, respectively, of the functional J_{f}(·) over the set T_{M} := {μ_{Q}: Q ∈ M} ⊆ S. When the function g satisfies the finiteness condition (1.4), the same is true for the function f defined in (1.6). In this case, by the relative compactness of the set T_{M}, it is possible to find a sequence {Q^{n}} in M and a measure μ^{∗} = μ^{∗}(f) ∈ S such that μ_{Q^{n}} ⇒ μ^{∗} and lim_{n} J_{f}(μ_{Q^{n}}) = ℂ^{∗}. By Proposition 1.1 we have ℂ^{∗} = J_{f}(μ^{∗}). Similarly, there is a (possibly different) sequence {Q^{n}} in M and a measure μ_{∗} = μ_{∗}(f) ∈ S such that μ_{Q^{n}} ⇒ μ_{∗} and lim_{n} J_{f}(μ_{Q^{n}}) = ℂ_{∗}. By Proposition 1.1 we have ℂ_{∗} = J_{f}(μ_{∗}). Of course, in principle one cannot choose the measures μ^{∗} and μ_{∗} independently of f. However, we restrict our attention to the case where this can be done. As a justification, we mention that many models of interest have this property. In particular, such are the models considered in the works cited above. The next two propositions describe the situation where the measures μ^{∗} and μ_{∗} can be chosen independently of f.

Proposition 1.3 Let M ≠ ∅ and μ ∈ S be such that μ_{Q} ≼ μ for any Q ∈ M. There is equivalence between:

(a) sup_{M} J_{f}(μ_{Q}) = J_{f}(μ) for all f ∈ C;

(b) there is a strictly convex function f ∈ C satisfying (1.4) such that sup_{M} J_{f}(μ_{Q}) = J_{f}(μ);

(c) there is a sequence {Q^{n}} in M such that μ_{Q^{n}} ⇒ μ.

Moreover, if these conditions are satisfied then μ = sup {μ_{Q}: Q ∈ M}.

Proposition 1.4 Let M ≠ ∅ and μ ∈ S be such that μ ≼ μ_{Q} for any Q ∈ M. There is equivalence between:

(a) inf_{M} J_{f}(μ_{Q}) = J_{f}(μ) for all f ∈ C satisfying (1.4);

(b) there is a strictly convex function f ∈ C satisfying (1.4) such that inf_{M} J_{f}(μ_{Q}) = J_{f}(μ);

(c) there is a sequence {Q^{n}} in M such that μ_{Q^{n}} ⇒ μ.

Moreover, if these conditions are satisfied then μ = inf {μ_{Q}: Q ∈ M}.

Proof of Proposition 1.3. Since (a)⇒(b) is evident, (c)⇒(a) follows from Proposition 1.1, and the last statement follows from Proposition 1.2(a), it remains to prove (b)⇒(c). Due to compactness arguments as above, there exist a sequence {Q^{n}} in M and a measure μ′ ∈ S such that μ_{Q^{n}} ⇒ μ′ and J_{f}(μ′) = sup_{M} J_{f}(μ_{Q}) = J_{f}(μ). By Proposition 1.1, we have μ′ ≼ μ. Using Proposition 1.2(b) and conditional Jensen’s inequality, we obtain from the strong convexity of f that μ′ = μ.

Proof of Proposition 1.4 follows the same lines as the previous one.

If the conditions (a)–(c) of Proposition 1.3 are not satisfied, then, by Proposition 1.2(c), there is still the measure μ^{0} = sup {μ_{Q}: Q ∈ M}, but we only have the inequality sup_{M} J_{f}(μ_{Q}) ≤ J_{f}(μ^{0}), and the inequality is strict for strictly convex f satisfying (1.4). In the case of lower bounds the situation is similar.

1.4. Propositions 1.3 and 1.4 suggest what to do in order to find the upper and lower prices:

1) find a measure μ^{∗} (resp. μ_{∗}) which is an upper (resp. lower) bound for the set {μ_{Q}: Q ∈ M}, i.e. μ_{Q} ≼ μ^{∗} (resp. μ_{∗} ≼ μ_{Q}) for all Q ∈ M;

2) find a sequence {Q^{n}} in M such that μ_{Q^{n}} ⇒ μ^{∗} (resp. μ_{Q^{n}} ⇒ μ_{∗}).

Then

ℂ^{∗} = J_{f}(μ^{∗}) (resp. ℂ_{∗} = J_{f}(μ_{∗})).

If only the first step can be realized then we obtain an estimate from above for the upper price or an estimate from below for the lower price. Of course, these two steps, in one or another form, are presented at the papers where the upper and lower prices are found. It is useful to remark that the measure μ_{∗} = inf {μ_{Q}: Q ∈ M} always satisfies the properties μ_{∗}({0}) = 0 and ∫ x μ_{∗}(dx) = 1 by Proposition 1.2(b). Thus, it is not surprising that one can often find a probability measure Q_{∗} on (Ω, F) such that μ_{∗} = L(Z_{T} | Q_{∗}), and then we have ℂ_{∗} = E_{Q_{∗}}f(Z_{T}). Moreover, it often happens that Z is a martingale under Q_{∗} and there are no other measures equivalent to Q_{∗} for which Z is a martingale. This means that we have a complete arbitrage-free market model under Q_{∗} and ℂ_{∗} is the just the fair price of the option in that model. In the case of the upper price the situation may be different. It may happen that, for μ^{∗} = sup {μ_{Q}: Q ∈ M}, we have μ^{∗}({0}) > 0 (which does not change the situation drastically) or/and ∫ x μ^{∗}(dx) < 1. For example, a standard situation is μ^{∗} = δ_{{0}}. Even if μ^{∗} can be realized as μ^{∗} = L(Z_{T} | Q^{∗}) but if ∫ x μ^{∗}(dx) < 1, we do not have the property ℂ^{∗} = E_{Q^{∗}}f(Z_{T}) if \frac{f(∞)}{∞} ≠ 0, it may happen that the market corresponding to Q^{∗} admits arbitrage etc. A good illustration for the aforesaid is the model Z = exp(\overline{X}), where \overline{X} is a Lévy process, studied in detail by Jakubenas (1998) (we reserve the notation X for the stochastic logarithm of Z, see the subsections 1.5 and 3.1). He considers two cases: M = P and M = \{Q ∈ P : \overline{X} is a Lévy process under Q\}. As it follows from his Theorem 2, if P ≠ ∅, in both cases the least upper bounds μ^{∗} coincide, the greatest lower bounds μ_{∗} coincide, and conditions (a)–(c) of Propositions 1.3 and 1.4 are satisfied. Furthermore, μ_{∗} is the distribution of Z_{T} = exp(\overline{X}_{T}), where \overline{X} is one of the following Lévy processes: 1) \overline{X} ≡ 0; 2) \overline{X} = σW, where W is a standard Brownian motion, σ > 0; 3) \overline{X}_{t} = aπ_{t} + bt, where π is a Poisson process of intensity λ, a ≠ 0, and b is determined from the condition that exp(\overline{X}) is a martingale. In all the cases the corresponding market model is arbitrage-free and complete. The description of the upper bound is more complex. First, one possibility is μ^{∗} = δ_{{0}}. Second and third, μ^{∗} is represented through a Wiener process or a Poisson process, as happened to μ_{∗} in the cases 2) and 3) above. Fourth, μ^{∗} = δ_{{−aT }}, a > 0, and if one takes Z = exp (\overline{X}), \overline{X}_{t} = −at, then the market model admits arbitrage. Fifth, μ^{∗} can be represented as the distribution of Z_{T}, where Z is the stochastic exponential E(X), X is a Lévy process of the form X_{t} = −π_{t} + bt, where π is a Poisson process of intensity λ and b > 0 is determined from the condition that X is a martingale; in other words, Z_{t} = exp(bt) up to the first jump of π and Z_{t} = 0 after it. The corresponding market model is arbitrage-free and complete, but the price process may take zero value. Note that in all cases, excepting the first one for the upper bound, the process Z under the extremal measures can be also represented as Z = E(X), where X is a Lévy process (and a martingale excepting the fourth case for the upper measure). Jakubenas (1998) also founds the quantities sup {γ(Q) : Q ∈ P_{loc}} and inf {γ(Q) : Q ∈ P_{loc}\}. He considers only nonnegative f with f(0) = 0 and \frac{f(∞)}{∞} = 1, hence, as it was already mentioned, the upper bound is the same as in the case of equivalent martingale measures, while the lower bound is usually strictly less for P_{loc}. The explanation is easy: we still have μ_{∗} ≼ μ_{Q} for any Q ∈ P_{loc} (which is natural in view of our results in Section 5) so that inf {J_{f}(μ_{Q}): Q ∈ P_{loc}} = J_{f}(μ_{∗}), but, if Q ∈ P_{loc} \ P and \frac{f(∞)}{∞} > 0 then γ(Q) < J_{f}(μ_{Q}), see (1.3).

1.5. In the previous subsection we formulated two steps that say what to do in order to find the upper and lower prices. The main subject of our paper deals with results concerning how to perform these two steps for different market models. We start explaining the terms in which our main results are formulated. It turns out to be most convenient to take the stochastic logarithm X of Z and to express the conditions in terms of the triplet T_{Q} of local characteristics of X under martingale measures Q. There are different reasons for this. A suitable description of the class {μ_{Q}: Q ∈ M} is not available even for simple models (of course, if the market is not complete). What we know is Girsanov’s transformation rule connecting T_{Q} for an arbitrary Q ∼ P and the triplet T_{P} of X corresponding to the original measure P; this also allows us to construct measures Q with desired properties of T_{Q}. The property of X to be a Q-local martingale (which is equivalent to Q ∈ P_{loc}) is easily expressed in terms of T_{Q}. Next, it is convenient to consider the process Z under Q ∈ P as the density process of the probability measure \overline{Q} defined by d\overline{Q} = Z_{T} dQ with respect to Q. Then we can regard the weak convergence of measures in the second step as the weak convergence of the likelihoods and to use corresponding limit theorems, whose assumptions are expressed e.g. in terms of Hellinger processes, which in its turn have a simple representation through T_{Q}. Finally, our comparison result in Section 5 has a simple form in the same terms. As a tool for the first step of our approach we prove a comparison result in Section 5, which we call Comparison Lemma. It gives sufficient conditions under which the distributions L(Z_{T} | Q) and L(Z_{T}^{∗} | Q^{∗}), where Z is a nonnegative Q-supermartingale and Z^{∗} is a nonnegative Q^{∗}-supermartingale (defined maybe on different stochastic bases) are comparable in the sense of the partial ordering ≼. In the option pricing context L(Z_{T}^{∗} | Q^{∗}) plays the role of μ^{∗} or μ_{∗}, while Z and Q are interpreted as before. The conditions on Z^{∗} are restrictive, in particular, it is assumed that Z^{∗} has a Markovian structure. The result generalizes the arguments used in Bellamy and Jeanblanc (2000) and in Jakubenas (1998), see also El Karoui et al. (1998). As far as we know, this method has never been used for comparison of experiments in the statistical literature. In subsection 5.2 we check that our assumptions on Z^{∗} are satisfied if Z^{∗} = E(X^{∗}), where X^{∗} is a process with independent increments, or Z^{∗} is a diffusion. In subsection 5.3 we indicate some conditions in terms of T_{P} guaranteeing that the assumptions of the Comparison Lemma are satisfied for any Q ∈ P. Tools for the second step are considered in Sections 4 and 6. In Section 4 we deal with the special case μ^{∗} = δ_{{0}} or μ_{∗} = δ_{{1}}. Then conditions (c) of Propositions 1.3 and 1.4 are equivalent to the convergence of the variation distance ∥Q^{n} − \overline{Q^{n}}∥ to 2 and 0 respectively, where d\overline{Q^{n}} = Z_{T} dQ^{n}. Thus we can use predictable criteria for the convergence in variation and for the asymptotic separability of measures, see e.g. Chapter V in Jacod and Shiryaev (1987). At the final stage of the preparation of this paper we got acquainted with an unpublished manuscript by Jacod (1997), where essentially the same results included in our Propositions 4.1 and 4.2 were proved. Similar propositions when Z is a continuous process can be found in Frey and Sin (1999). If μ_{∗} = δ_{{0}} or μ_{∗} = δ_{{1}}, under conditions (a)–(c) of Propositions 1.3 and 1.4 we have

ℂ^{∗} = J_{f}(δ_{\{0\}}) = f(0) + \frac{f(∞)}{∞} = B_{T}^{−1}g(0) + B_{0}^{−1}S_{0} \frac{g(∞)}{∞}

and

ℂ_{∗} = J_{f}(δ_{{1}}) = f(1) = B_{T}^{−1}g(B_{0}^{−1}S_{0}B_{T})

respectively. Note that, in the general case, by Proposition 1.2(d) we always have these bounds as an upper estimate for ℂ^{∗} and lower estimate for ℂ_{∗}, i.e.

B_{T}^{−1}g(B_{0}^{−1}S_{0}B_{T}) ≤ ℂ_{∗} ≤ ℂ^{∗} ≤ B_{T}^{−1}g(0) + B_{0}^{−1}S_{0} \frac{g(∞)}{∞}.  (1.7)

Of course, these inequalities can be obtained directly: the first one follows from Jensen’s inequality and the second one follows from the inequality g(x) ≤ g(0) + x\frac{g(∞)}{∞}, x ∈ ℝ_{+}. We call these bounds universal bounds. The fact that there exist such universal bounds for any market model of the considered type was stated by Eberlein and Jacod (1997). In Section 6 we are interested how to perform the second step if the measure μ does not coincide with δ_{{0}} or δ_{{1}}. Unlike limit theorems for likelihood processes, our task is more simple since the process Z is a strictly positive martingale with respect to any Q^{n} ∈ M, which means that it is the density process for equivalent measures. However, the existing results on this subject cannot be completely adapted to our setting, see the discussion in Section 6 and the references therein. We prove two new limit theorems. The first one deals with the case where the limit is the distribution of Z_{T}^{∗}, Z^{∗} = E(X^{∗}) is a nonnegative supermartingale, and X^{∗} is a Lévy process. The second theorem corresponds to the case where Z^{∗} is a diffusion. The rest of the paper is as follows. In Section 2 a short review of comparison and convergence of binary statistical experiments is given. Section 3 contains information about martingale measures and Hellinger processes. In Section 7 we examine the bounds of prices for a general random walk market model, and Section 8 carries the same task when the underlying process is the solution of a stochastic differential equation with jumps. Some concluding remarks are presented in Section 9.

2 Binary statistical experiments

Here we give a brief review of the theory of comparison and convergence of binary statistical experiments. All the statements are known and may be found e.g. in Strasser (1985), Torgersen (1991), Shiryaev and Spokoiny (2000). We also refer to Liese and Vajda (1987) for the notion of f-divergence and its properties. The notation introduced in subsection 1.2 is used.

2.1. By a binary experiment we mean a collection 𝔼 = (Ω, F, (P, P′)), where (Ω, F) is a measurable space and (P, P′) is an ordered pair of probability measures on (Ω, F). In the sequel we consider only binary statistical experiments, so the word ‘binary’ will be omitted. The experiment 𝔼 is homogeneous if P and P′ are equivalent. If P = P′ then 𝔼 is said to be a totally non informative experiment. If the measures P and P′ are singular then 𝔼 is said to be a totally informative experiment. Let Z be a random variable on (Ω, F) with values in [0, +∞] such that

P′(A) = ∫_{A} Z dP + P′(A ∩ {Z = ∞}) for any A ∈ F.

Given P and P′ the variable Z always exists, is unique up to (P +P′)-null sets, and is called the generalized density of P′ with respect to P. We have P (Z < ∞) = P′(Z > 0) = 1. If a σ-finite measure λ on (Ω, F) dominates both P and P′, z = dP/dλ and z′ = dP′/dλ are the corresponding Radon–Nikodym densities, then Z = z′/z P - and P′-a.s. (where 0/0 is interpreted arbitrarily, say, 0/0 = 0). Of course, if P′ is absolutely continuous with respect to P then Z is just the Radon–Nikodym density of P′ with respect to P. In general, a (P -a.s. finite) random variable Z is the density of the absolutely continuous part of P′ with respect to P. We associate the following objects with an experiment 𝔼 = (Ω, F, (P, P′)).

1. A probability measure μ_{𝔼} belonging to S defined by

μ_{𝔼}(B) = P (Z ∈ B), B ∈ B(ℝ_{+}).

Conversely, for any probability measure μ ∈ S there is an experiment 𝔼 such that μ = μ_{𝔼}. For example, take Ω = [0, +∞], F = B([0, ∞]), P = μ, P′(dx) = xμ(dx) + (1 − ∫_{ℝ_{+}} xμ(dx)) δ_{{+∞}}(dx).

2. A probability measure ν_{𝔼} on ([0, 2], B([0, 2])) defined by

ν_{𝔼}(B) = Q(z ∈ B), B ∈ B([0, 2]),

where Q = (P +P′)/2, z = dP/dQ. The measure ν_{𝔼} has the property ∫_{[0,2]} xν_{𝔼}(dx) = 1. Moreover, for any probability measure ν on ([0, 2], B([0, 2])) with ∫_{ℝ_{+}} xν(dx) = 1 there is an experiment 𝔼 such that ν = ν_{𝔼}. For example, take Ω = [0, 2], F = B([0, 2]), P (dx) = xν(dx), P′(dx) = (2 − x)ν(dx).

3. A function β_{𝔼}(α), α ∈ [0, 1], where β_{𝔼}(α) is the power of the most powerful level α test for testing ‘P ’ against ‘P′’, i.e.

β_{𝔼}(α) = sup_{φ∈Φ:Eφ≤α} E′φ,

where Φ is the set of all measurable functions φ on (Ω, F) with values in [0, 1] (test functions), E and E′ are expectations with respect to P and P′ respectively. The function β_{𝔼}(α) is a nondecreasing continuous concave function with values in [0, 1] and β_{𝔼}(1) = 1. Conversely, if β(α) (α ∈ [0, 1]) is a nondecreasing continuous concave function with values in [0, 1] satisfying β(1) = 1, then β = β_{𝔼}, where (Ω, F) = ([0, 1], B([0, 1])), P is the Lebesgue measure, P′(dα) = dβ(α) + β(0)δ_{{0}}(dα), and 𝔼 = (Ω, F, (P, P′)).

4. A function b_{𝔼}(π), π ∈ [0, 1], where b_{𝔼}(π) is the minimum Bayes risk for testing ‘P ’ against ‘P′’ with a priori probabilities 1 − π and π, i.e.

b_{𝔼}(π) = inf_{φ∈Φ}{(1 − π)Eφ + πE′(1 − φ)}.

The function b_{𝔼}(π) is a concave function and 0 ≤ b_{𝔼}(π) ≤ min{π, 1 − π}, π ∈ [0, 1]. Moreover, if b(π) is a concave function satisfying 0 ≤ b(π) ≤ min{π, 1−π}, π ∈ [0, 1], then there is an experiment 𝔼 such that b = b_{𝔼}. Note that functions from this class are uniformly equicontinuous. Note also that the distance in variation between P and P′ defined as ∥P − P′∥ = 2 sup_{A∈F} |P (A) − P′(A)| satisfies

∥P − P′∥ = 2(1 − 2b_{𝔼}(1/2)).  (2.1)

5. A function H_{𝔼}(α), α ∈ (0, 1), where H_{𝔼}(α) is the Hellinger integral of order α for P and P′:

H_{𝔼}(α) = H(α; P, P′) = EZ^{1−α}.

2.2. In the following definition λ is a σ-finite measure dominating P and P′, z and z′ are the densities of P and P′ respectively with respect to λ. The conventions 0f(\frac{a}{0}) = a\frac{f(∞)}{∞} are used.

Definition 1 (Csiszár (1963)) Let f ∈ C. The quantity

J_{f}(P′, P) = ∫ zf(\frac{z′}{z} )dλ

is called the f-divergence of P′ and P.

It is easy to check that the definition is correct and does not depend on the choice of λ. We shall also write J_{f}(𝔼) instead of J_{f}(P′, P), cf. (2.2) below. If 𝔼 is homogeneous then J_{f}(𝔼) = Ef(Z). The same is true if f is nonnegative and decreasing.

Remark 2.1 If f(x) = ax + b, x > 0, a, b ∈ ℝ, then J_{f}(𝔼) = a + b does not depend on 𝔼. Due to this fact, one is free to subtract a linear function from an f ∈ C to obtain additional desired properties of f, e.g. f ≥ 0.

It is easy to express J_{f}(𝔼) in terms of μ_{𝔼} and ν_{𝔼}, explaining in particular the notation in (1.3). Namely,

J_{f}(𝔼) = J_{f}(μ_{𝔼}) = ∫_{[0,2]} f̃(x)ν_{𝔼}(dx),  (2.2)

where f̃(x), x ∈ [0, 2], is a continuous convex function defined by

f̃(x) = \{ \begin{matrix}2\frac{f(∞)}{∞}, & if & x = 0, \\ xf(\frac{2−x}{x}), & if & 0 < x < 2, \\ 2f(0), & if & x = 2. \end{matrix}

Some of the quantities introduced in 2.1 can be expressed via f-divergences. In particular, put f_{π}(x) = max{1 − π − πx, 0}, π ∈ [0, 1], g_{1}(x) = |x − 1|, g_{2}(x) = 2(1 − x)^{+}, g_{3}(x) = 2(x − 1)^{+}, and φ_{α}(x) = αx − x^{1−α} + 1 − α, α ∈ (0, 1). Then

b_{𝔼}(π) = 1 − π − J_{f_{π}}(𝔼),  (2.3)
∥P − P′∥ = J_{g_{i}}(𝔼) (i = 1, 2, 3),
H_{𝔼}(α) = 1 − J_{φ_{α}}(𝔼).  (2.4)

2.3. The next definition introduces a partial pre-ordering for experiments, that motivated our partial ordering in Proposition 1.2.

Definition 2 An experiment 𝔼 is said to be more informative than an experiment 𝔼̃ (denoted 𝔼 ≽ 𝔼̃ or 𝔼̃ ≼ 𝔼) if β_{𝔼}(α) ≥ β𝔼̃(α) for all α ∈ [0, 1]. The experiments 𝔼 and 𝔼̃ are said to be equivalent (denoted 𝔼 ∼ 𝔼̃) if 𝔼 ≼ 𝔼̃ and 𝔼̃ ≼ 𝔼, i.e. if β_{𝔼}(α) = β𝔼̃(α) for all α ∈ [0, 1]. Equivalent experiments are said to have the same type, so a type is a collection of equivalent experiments.

Simple considerations based on the Neyman–Pearson lemma lead to the following result.

Proposition 2.1 𝔼 ≽ 𝔼̃ if and only if b_{𝔼}(π) ≤ b𝔼̃(π) for all π ∈ [0, 1]. In particular, 𝔼 ∼ 𝔼̃ if and only if b_{𝔼}(π) = b𝔼̃(π) for all π ∈ [0, 1].

The next statement gives a comparison criterion in terms of measures μ_{𝔼}. In fact, it is an almost immediate consequence of the previous proposition.

Proposition 2.2 𝔼 ≽ 𝔼̃ if and only if

∫_{0}^{a} μ_{𝔼}([0, x]) dx ≥ ∫_{0}^{a} μ𝔼̃([0, x]) dx for all a ≥ 0.

In particular, 𝔼 ∼ 𝔼̃ if and only if μ_{𝔼} = μ𝔼̃.

It follows from Proposition 2.1 and (2.3) that, cf. (1.5),

𝔼 ≽ 𝔼̃ if and only if J_{f_{π}}(𝔼) ≥ J_{f_{π}}(𝔼̃), π ∈ [0, 1].  (2.5)

Using (2.5) and Remark 2.1, one easily shows that 𝔼 ≽ 𝔼̃ implies J_{f}(𝔼) ≥ J_{f}(𝔼̃) for any f which is the maximum of a finite set of linear functions on (0, +∞). This can be extended to all f ∈ C by using of monotone approximations.

Proposition 2.3 If 𝔼 ≽ 𝔼̃ then J_{f}(𝔼) ≥ J_{f}(𝔼̃) for any f ∈ C.

Another proof of this result is based on Proposition 2.4 below. Let C_{0}^{2} be the subset of C consisting of all bounded nonnegative decreasing twice continuously differentiable convex functions f with a compact support. Since any function f_{π}, π ∈ [0, 1], can be uniformly approximated by functions from C_{0}^{2}, we obtain the following result useful in applications.

Corollary 2.1 𝔼 ≽ 𝔼̃ if and only if ∫ f(x) μ_{𝔼}(dx) ≥ ∫ f(x) μ𝔼̃(dx) for any f ∈ C_{0}^{2}.

A standard tool for comparison of experiments is the randomization criterion. Proposition 1.2(b) is its corollary.

Proposition 2.4 Let 𝔼 = (Ω, F, (P, P′)) and 𝔼̃ = (Ω̃, F̃, (P̃, P̃′)) be experiments. If (Ω̃, F̃) is isomorphic to a Borel subset of a complete separable metric space equipped with the Borel σ-algebra, then 𝔼 ≽ 𝔼̃ if and only if there is a Markov kernel K from (Ω, F) to (Ω̃, F̃) such that P̃ = KP and P̃′ = KP′, where KP (A) := ∫_{Ω} K(ω, A)P (dω), A ∈ F̃, and KP′ is defined similarly.

Note that ‘if’ part of Proposition 2.4 is true without any assumptions on (Ω̃, F̃). We do not give here a comparison criterion in terms of measures ν_{𝔼}. Let us only mention that 𝔼 ∼ 𝔼̃ if and only if ν_{𝔼} = ν𝔼̃. A concluding remark is that 𝔼 ≽ 𝔼̃ implies H_{𝔼}(α) ≤ H𝔼̃(α) for any α ∈ (0, 1) (apply Proposition 2.3 and (2.4)). It is important to remark that the converse of this statement is not true. Nevertheless, H_{𝔼}(α) = H𝔼̃(α) for all α ∈ (0, 1) implies 𝔼 ∼ 𝔼̃.

2.4. Let (𝔼_{λ})_{λ∈Λ} be any collection of experiments. Since inf_{λ∈Λ} b_{𝔼_{λ}}(π) is a concave function in π, there is an experiment 𝔼^{∗} such that b_{𝔼^{∗}}(π) = inf_{λ∈Λ} b_{𝔼_{λ}}(π), π ∈ [0, 1]. Any such experiment 𝔼^{∗} is a least upper bound for (𝔼_{λ})_{λ∈Λ}. Note that β_{𝔼^{∗}} is the concave envelope of the pointwise supremum of β_{𝔼_{λ}}, λ ∈ Λ. Similarly, inf_{λ∈Λ} β_{𝔼_{λ}}(α) is a nondecreasing concave function in α. Hence, there is an experiment 𝔼_{∗} such that β_{𝔼_{∗}}(α) = inf_{λ∈Λ} β_{𝔼_{λ}}(α), α ∈ [0, 1]. Any such experiment 𝔼_{∗} is a greatest lower bound for (𝔼_{λ})_{λ∈Λ}. Note that b_{𝔼_{∗}} is the concave envelope of the pointwise supremum of b_{𝔼_{λ}}, λ ∈ Λ. If 𝔼 is an homogeneous experiment and 𝔼̃ ≼ 𝔼 then 𝔼̃ is an homogeneous experiment too. Thus, a greatest lower bound for (𝔼_{λ})_{λ∈Λ} is a homogeneous experiment if at least one of 𝔼_{λ} is homogeneous. A least upper bound for (𝔼_{λ})_{λ∈Λ} need not be a homogeneous experiment even if all 𝔼_{λ} are homogeneous.

2.5. To introduce the notion of weak convergence of experiments, we start with a preliminary definition of the deficiency of one experiment with respect to another one. Usually, the deficiency is defined in terms of errors corresponding to test functions. Our definition is equivalent to it, see e.g. Strasser (1985, p. 76).

Definition 3 A deficiency of an experiment 𝔼 with respect to an experiment 𝔼̃ is the quantity

δ_{2}(𝔼, 𝔼̃) = 2 sup_{π∈[0,1]}[b_{𝔼}(π) − b𝔼̃(π)].

The deficiency distance between 𝔼 and 𝔼̃ is the quantity

Δ_{2}(𝔼, 𝔼̃) = max {δ_{2}(𝔼, 𝔼̃), δ_{2}(𝔼̃, 𝔼)} = 2 sup_{π∈[0,1]} |b_{𝔼}(π) − b𝔼̃(π)|.

Evidently, Δ_{2} satisfies the triangle inequality. Hence, strictly speaking, Δ_{2} is a pseudo-distance on the family of all experiments. Due to Proposition 2.1, 𝔼 ≽ 𝔼̃ if and only if δ_{2}(𝔼, 𝔼̃) = 0; 𝔼 ∼ 𝔼̃ if and only if Δ_{2}(𝔼, 𝔼̃) = 0. Hence, Δ_{2} defines a metric on the space of all types of experiments. Moreover, this metric space is a compact. The convergence in this space is called the weak convergence.

Definition 4 A sequence of experiments 𝔼_{n} converges weakly to an experiment 𝔼 (denoted 𝔼_{n} ⟶^{w} 𝔼) if Δ_{2}(𝔼_{n}, 𝔼) → 0, n → ∞.

Here we do not need expressions for deficiencies and deficiency distances in terms of measures μ_{𝔼} or functions β_{𝔼}. What is important for us are criteria for weak convergence of experiments.

Proposition 2.5 Assume that 𝔼_{n}, n = 1, 2, …, and 𝔼 are experiments. The following statements are equivalent:

(i) 𝔼_{n} ⟶^{w} 𝔼;

(ii) μ_{𝔼_{n}} ⇒ μ_{𝔼};

(iii) ν_{𝔼_{n}} ⇒ ν_{𝔼};

(iv) β_{𝔼_{n}}(α) converges pointwise on (0, 1] to β_{𝔼}(α);

(v) b_{𝔼_{n}}(π) converges uniformly (or pointwise) on [0, 1] to b_{𝔼}(π);

(vi) H_{𝔼_{n}}(α) converges pointwise on (0, 1) to H_{𝔼}(α).

Combining implication (i)⇒(iii), (2.2) and Fatou’s lemma or Lebesgue’s theorem on dominated convergence (considering nonnegative f due to Remark 2.1), we get the following key result.

Proposition 2.6 Assume that a sequence of experiments 𝔼_{n} converges weakly to an experiment 𝔼. If f ∈ C then

J_{f}(𝔼) ≤ lim inf_{n→∞} J_{f}(𝔼_{n}).

If, moreover, f(0) + \frac{f(∞)}{∞} < ∞ then

J_{f}(𝔼) = lim_{n→∞} J_{f}(𝔼_{n}).

2.6. Let 𝔼_{a} be a totally informative experiment and 𝔼_{i} a totally non informative experiment. The following facts are easy to verify:

• μ_{𝔼_{a}} = δ_{\{0\}}, b_{𝔼_{a}}(π) ≡ 0, H_{𝔼_{a}}(α) ≡ 0 (0 < α < 1), J_{f}(𝔼_{a}) = f(0) + \frac{f(∞)}{∞}, (f ∈ C),

• μ_{𝔼_{i}} = δ_{{1}}, b_{𝔼_{i}}(π) ≡ min{π, 1 − π}, H_{𝔼_{a}}(α) ≡ 1, (0 < α < 1), J_{f}(𝔼_{i}) = f(1) (f ∈ C).

Moreover, H_{𝔼}(1/2) = 0 implies 𝔼 ∼ 𝔼_{a} and H_{𝔼}(1/2) = 1 implies 𝔼 ∼ 𝔼_{i}. As a consequence of preceding results, we get the following statements.

Proposition 2.7 For any experiment 𝔼 and any f ∈ C

f(1) ≤ J_{f}(𝔼) ≤ f(0) + \frac{f(∞)}{∞}.

Remark 2.2 Compare the statement of the previous Proposition with the universal bounds in (1.7).

Proposition 2.8 Assume that 𝔼_{n} = (Ω_{n}, F_{n}, (P_{n}, P′_{n})), n = 1, 2, …, are experiments. The following statements are equivalent:

(i) 𝔼_{n} ⟶^{w} 𝔼_{a};

(ii) μ_{𝔼_{n}} ⇒ δ_{{0}};

(iii) lim_{n→∞} ∥P_{n} − P′_{n}∥ = 2;

(iv) lim_{n→∞} H_{𝔼_{n}}(1/2) = 0.

Proposition 2.9 Assume that 𝔼_{n} = (Ω_{n}, F_{n}, (P_{n}, P′_{n})), n = 1, 2, …, are experiments. The following statements are equivalent:

(i) 𝔼_{n} ⟶^{w} 𝔼_{i};

(ii) μ_{𝔼_{n}} ⇒ δ_{{1}};

(iii) lim_{n→∞} ∥P_{n} − P′_{n}∥ = 0;

(iv) lim_{n→∞} H_{𝔼_{n}}(1/2) = 1.

3 Martingale measures and Hellinger processes

In this section we recall some useful facts from the theory of martingales for reference purposes. The notation and the details can be found e.g. in Jacod and Shiryaev (1987).

3.1. Let Z = {Z_{t}}_{0≤t≤T} be a strictly positive semimartingale on a stochastic basis B = (Ω, F_{T} , {F_{t}}_{0≤t≤T} , P) with Z_{0} = 1. There are two convenient representations of Z either as the exponential Z = exp(\overline{X}) of \overline{X} = \{\overline{X}_{t}\}_{0≤t≤T} or as the stochastic exponential Z = E(X) of X = \{X_{t}\}_{0≤t≤T} with X_{0} = 0, the latter means that Z satisfies the stochastic differential equation

dZ_{t} = Z_{t−}dX_{t},  0 ≤ t ≤ T,  Z_{0} = 1,

the solution being written explicitly

Z_{t} = exp(X_{t} − \frac{1}{2}⟨X^{c}, X^{c}⟩_{t}) ∏_{0<s≤t}(1 + ΔX_{s})e^{−ΔX_{s}}.  (3.1)

Of course, X can be easily expressed through \overline{X} and vice versa. In particular, the jumps of X and \overline{X} are connected by the relation

ΔX = exp(Δ\overline{X}) − 1.  (3.2)

Some of the subsequent results have a more simple formulation in terms of X. This is why we use a more simple notation for X rather than \overline{X}. See anyway the discussion in Shiryaev (1999).

Let (B(ϕ), C, ν) and (\overline{B}(ϕ), \overline{C}, \overline{ν}) be the triplets of local characteristics of X and \overline{X} respectively with respect to a truncation function ϕ(x). Then up to a P -null set

\{ \begin{matrix}\overline{B}(ϕ) = B(ϕ) − \frac{1}{2}C + \{ϕ(log(1 + x)) − ϕ(x)\} ∗ ν, \\ \overline{C} = C, \\ g(ω, t, x) ∗ \overline{ν} = g(ω, t, log(1 + x)) ∗ ν. \end{matrix}  (3.3)

Note that ΔX > −1, hence ν charges only the set {(ω, t, x): 0 < t ≤ T, x ∈ (−1, 0) ∪ (0, ∞)}.

3.2. Let Q be another probability measure on (Ω, F_{T}) equivalent to P. By (3.1) the representation Z = E(X) also holds with respect to Q. Let T_{Q} = (B^{Q}(ϕ), C^{Q}, ν^{Q}) be the triplet of local characteristics of X relative to Q. By Girsanov’s Theorem for semimartingales, see Proposition III.3.24 in Jacod and Shiryaev (1987), there exist a predictable function β^{Q} = {β_{t}^{Q}}_{0≤t≤T} and a strictly positive function Y^{Q} = Y^{Q}(ω, t, x) measurable with respect to the predictable σ-algebra on Ω × [0, T] × ℝ such that P -a.s.

|β^{Q}| · C_{T} < ∞, |ϕ(x)(Y^{Q} − 1)| ∗ ν_{T} < ∞,  (3.4)

and

\{ \begin{matrix}B^{Q}(ϕ) = B(ϕ) + β^{Q} · C + ϕ(x)(Y^{Q} − 1) ∗ ν, \\ C^{Q} = C, \\ ν^{Q} = Y^{Q} · ν. \end{matrix}  (3.5)

Now assume that Q ∈ P, i.e. Q ∼ P and Z is a Q-martingale. Then X is a Q-local martingale. By Proposition II.2.29 in Jacod and Shiryaev (1987), Q-a.s.

(x^{2} ∧ |x|) ∗ ν_{T}^{Q} < ∞  (3.6)

and

B^{Q}(ϕ) + (x − ϕ(x)) ∗ ν^{Q} = 0.  (3.7)

Note that condition (3.7) implies that Z is a Q-local martingale but it is not sufficient for Z to be a Q-martingale. In order to verify that Z = E(X) is an uniformly integrable martingale a sufficient condition is the existence of a constant H such that

⟨X^{c}, X^{c}⟩_{T} + \frac{x^{2}}{1 + |x|} ∗ ν_{T}^{Q} ≤ H.  (3.8)

See Theorem 12 in Kabanov et al. (1979).

3.3. Let again Q ∈ P. Define a probability measure \overline{Q} on F_{T} by d\overline{Q} = Z_{T} dQ. Then Z = \{Z_{t}\}_{0≤t≤T} is the density process of \overline{Q} with respect to Q. By Corollary IV.1.37 in Jacod and Shiryaev (1987), the Hellinger process h(α) = h(α; Q, \overline{Q}) of order α ∈ (0, 1) for Q and \overline{Q} has the form

h(α) = \frac{α(1 − α)}{2}C^{Q} + \{α + (1 − α)(1 + x) − (1 + x)^{1−α}\} ∗ ν^{Q}.  (3.9)

In particular,

h(\frac{1}{2}) = \frac{1}{8}C^{Q} + \frac{1}{2}(1 − \sqrt{1 + x})^{2} ∗ ν^{Q}.  (3.10)

3.4. By reasons that will be apparent later, let us consider a more general situation. Namely, assume that (Ω, F_{T} , {F_{t}}_{0≤t≤T} , Q) is a stochastic basis, Z = {Z_{t}}_{0≤t≤T} is a nonnegative supermartingale defined on this basis, and Z is representable in the form

Z = E(X)  (3.11)

(note that in general, there are nonnegative supermartingales Z with Z_{0} = 1 that cannot be represented in this form). The process X = {X_{t}}_{0≤t≤T} is not uniquely determined by (3.11) in general. Nevertheless, one can choose X such that X_{0} = 0 and X is a Q-local supermartingale with ΔX ≥ −1. For example, one can take X = \frac{1_{\{Z_{−}>0\}}}{Z_{−}} · Z. We shall assume that X satisfies these assumptions. Denote by A the predictable non-increasing process in the Doob–Meyer decomposition of X, i.e. A_{0} = 0 and X − A is a Q-local martingale. Let (B(ϕ), C, ν) be the triplet of local characteristics of X with respect to Q (we omit Q in the indices here). Of course, ν charges only the set {(ω, t, x): 0 < t ≤ T, x ∈ [−1, 0) ∪ (0, ∞)}. By Proposition II.2.29 in Jacod and Shiryaev (1987) we have that Q-a.s.

(x^{2} ∧ |x|) ∗ ν_{T} < ∞  (3.12)

and the processes A and B(ϕ) are connected by the relation

A = B(ϕ) + (x − ϕ(x)) ∗ ν.  (3.13)

Conversely, if X is a Q-local supermartingale with X_{0} = 0 and ΔX ≥ −1, then Z = E(X) is a nonnegative Q-local supermartingale, hence a Q-supermartingale.

4 Attainability of universal bounds

In this section we focus on the problem of the attainability of universal bounds, giving, in particular, criteria in terms of Hellinger processes, i.e. predictable criteria. Our setting is the same as in Section 1. In particular, B, S and g satisfy the assumptions of subsection 1.1, f and Z are defined as in (1.6). It is always assumed that M ≠ ∅.

4.1. For any probability measure Q (without or with indices), Q ∈ M, let \overline{Q} (without or with the same indices) be the probability measure defined by d\overline{Q} = Z_{T} dQ. The expression for the Hellinger process for Q and \overline{Q} in terms of T_{Q} is given in subsection 3.3.

Proposition 4.1 (a) The following statements are equivalent:

(i) For any g ∈ C the upper universal bound is attained, i.e.

ℂ^{∗} = sup_{Q∈M} γ(Q) = B_{T}^{−1}g(0) + S_{0} \frac{g(∞)}{∞} = f(0) + \frac{f(∞)}{∞}.  (4.1)

(ii) There exists a sequence {Q^{n}}_{n≥1}, Q^{n} ∈ M, such that

lim_{n→∞} Q^{n}(Z_{T} > ε) = 0 for any ε > 0.

(ii′) There exists a sequence {Q^{n}}_{n≥1}, Q^{n} ∈ M, such that

lim_{n→∞} \overline{Q^{n}}(Z_{T} < N) = 0 for any N > 0.

(b) The following statement implies (i), (ii) and (ii′):

(iii) There exists a sequence {Q^{n}}_{n≥1}, Q^{n} ∈ M, such that

lim_{n→∞} Q^{n}(h_{T}^{n} > N) = 1 for any N > 0,

where h^{n} = h(1/2; Q^{n}, \overline{Q^{n}}) is the Hellinger process of order 1/2 for Q^{n} and \overline{Q^{n}}.

(c) If P (inf_{t≤T} \frac{Z_{t}}{Z_{t−}} ≤ β) = 0 for some β > 0 (in particular, if Z is P -a.s. continuous), then (iii) is equivalent to (i), (ii) and (ii′).

Proposition 4.2 The following statements are equivalent:

(i) For any g ∈ C with g(0) + \frac{g(∞)}{∞} < ∞ the lower universal bound is attained, i.e.

ℂ_{∗} = inf_{Q∈M} γ(Q) = B_{T}^{−1}g(S_{0}B_{T}) = f(1).  (4.2)

(ii) There exists a sequence {Q^{n}}_{n≥1}, Q^{n} ∈ M, such that

lim_{n→∞} Q^{n}(|Z_{T} − 1| > ε) = 0 for any ε > 0.

(iii) There exists a sequence {Q^{n}}_{n≥1}, Q^{n} ∈ M, such that

lim_{n→∞} Q^{n}(h_{T}^{n} > ε) = 0 for any ε > 0,

where h^{n} = h(1/2; Q^{n}, \overline{Q^{n}}) is the Hellinger process of order 1/2 for Q^{n} and \overline{Q^{n}}.

Proof of Proposition 4.1. (a) Applying (i) with f(x) = |x−1|, one can find a sequence {Q^{n}}_{n≥1}, Q^{n} ∈ M, such that

∥Q^{n} − \overline{Q^{n}}∥ → 2, n → ∞.  (4.3)

Due to Proposition 2.8 the last property is equivalent to (ii) or to

𝔼^{n} = (Ω, F_{T} , (Q^{n}, \overline{Q^{n}})) ⟶^{w} 𝔼_{a}, n → ∞.

Applying Proposition 2.8 to the experiments 𝔼̂^{n} = (Ω, F_{T} , (\overline{Q^{n}}, Q^{n})), we see that (4.3) is also equivalent to

lim_{n→∞} \overline{Q^{n}}(dQ^{n}/d\overline{Q^{n}} > ε) = lim_{n→∞} \overline{Q^{n}}(Z_{T} < ε^{−1}) = 0 for any ε > 0,

which is nothing else than (ii′). Conversely, if there is a sequence {Q^{n}}_{n≥1}, Q^{n} ∈ M, such that 𝔼^{n} ⟶^{w} 𝔼_{a}, n → ∞, then, by Proposition 2.6,

lim inf_{n→∞} γ(Q^{n}) = lim inf_{n→∞} J_{f}(𝔼^{n}) ≥ f(0) + \frac{f(∞)}{∞}

for any f ∈ C. Combining this statement with Proposition 2.7, we get (i).

(b) As Z_{0} = 1, the restrictions of Q and \overline{Q} onto the σ-algebra F_{0} coincide for any Q ∈ M. By part (ii) of Theorem V.4.32 in Jacod and Shiryaev (1987), the condition (iii) implies (4.3).

(c) It is enough to show that if (iii) is violated, then (4.3) is not valid for any sequence {Q^{n}}_{n≥1}, Q^{n} ∈ M. Assume the converse. Then there is a sequence {Q^{n}}_{n≥1}, Q^{n} ∈ M, satisfying (4.3) and such that

lim sup_{n→∞} Q^{n}(h_{T}^{n} > N) < 1 for some N > 0,

where h^{n} = h(1/2; Q^{n}, \overline{Q^{n}}). Applying part (a) of Theorem V.2.4 in Jacod and Shiryaev (1987) with P^{n} = \overline{Q^{n}} and P^{′n} = Q^{n}, we arrive at a contradiction. Indeed, since Q^{n} ∼ P, we have Q^{n}(inf_{t≤T} \frac{Z_{t}}{Z_{t−}} ≤ β) = 0, hence the process i^{n}(β) in that Theorem satisfies i^{n}(β) = 0, Q^{n}-a.s. for any n.

Proof of Proposition 4.2. Similar to the preceding proof (use Proposition 2.9 instead of Proposition 2.8), one shows that both (i) and (ii) are equivalent to the existence of a sequence {Q^{n}}_{n≥1}, Q^{n} ∈ M, such that

∥Q^{n} − \overline{Q^{n}}∥ → 0, n → ∞.  (4.4)

The equivalence of (4.4) and (iii) follows from part (i) of Theorem V.4.32 in Jacod and Shiryaev (1987).

Remark 4.1 It may happen that we have equality in (4.2) for a certain function g ∈ C, for example, for g(x) = (x−a)^{+} or g(x) = (a−x)^{+}, but not for all g ∈ C with g(0) + \frac{g(∞)}{∞} < ∞. However, slightly modifying the above proof, one can show that if the corresponding function f is strictly convex at point 1 in the sense that f does not coincide with a linear function on any interval (1 − δ, 1 + δ), δ > 0, then equality (4.2) for this f implies (4.4) and hence the statements (i)–(iii) of Proposition 4.2. An example of such a function g is g(x) = (x − B_{0}^{−1}S_{0}B_{T})^{+}, that according to (1.6) gives f(x) = B_{0}^{−1}S_{0}(x − 1)^{+}. Similarly, if equality (4.1) takes place for a function g ∈ C such that g(0)+ \frac{g(∞)}{∞} < ∞ and g is not identically linear on (0, ∞), then the statements (i), (ii) and (ii′) of Proposition 4.1 are valid.

Remark 4.2 Propositions 4.1 and 4.2 with the above remark contain as particular cases Propositions 2.3 and 2.4 in Frey and Sin (1999). In the general case similar results were proved in an unpublished manuscript by Jacod (1997).

4.2. The preceding results allow us to formulate sufficient conditions for the nonattainability of the universal bounds in the terms of the original measure P. For brevity, we shall say that the upper (resp. lower) universal bound is non-attainable if the statement (i) of Proposition 4.1 (resp. 4.2) does not hold with M = P (then it does not hold for any M ⊂ P). According to Remark 4.1 this means that

sup_{Q∈P} γ(Q) < B_{T}^{−1}g(0) + S_{0} \frac{g(∞)}{∞}

for any function g ∈ C such that g(0) + \frac{g(∞)}{∞} < ∞ and g is not identically linear on (0, ∞) (resp.

inf_{Q∈P} γ(Q) > B_{T}^{−1}g(S_{0}B_{T})

for any function g ∈ C which is strictly convex at point S_{0}B_{T}).

1) If there is a number δ > 0 such that P (Z_{T} < δ) = 0 then the upper universal bound is non-attainable. Indeed, then we have Q(Z_{T} < δ) = 0 for any Q ∈ P, therefore the statement (ii) of Proposition 4.1 does not hold.

2) Similarly, if there is a number N > 0 such that P (Z_{T} > N) = 0 then the upper universal bound is non-attainable (the statement (ii′) of Proposition 4.1 does not hold).

3) Similarly, if there is a number δ > 0 such that P (|Z_{T} − 1| < δ) = 0 then the lower universal bound is non-attainable (use implication (i)⇒(ii) in Proposition 4.2).

4) Now let X be a P -semimartingale such that Z = E(X), and let (B(ϕ), C, ν) be the triplet of local characteristics of X (relative to P) with respect to a truncation function ϕ(x). 4.1) If there is a number δ > 0 such that P (C_{T} < δ) = 0, then the lower universal bound is non-attainable. Indeed, for any Q ∈ P the Hellinger process h = h(1/2; Q, \overline{Q}) of order 1/2 for Q and \overline{Q} satisfies Q-a.s.

h_{T} ≥ \frac{1}{8}C_{T}

due to (3.10) and (3.5), and the statement follows from the implication (i)⇒(iii) in Proposition 4.2. 4.2) Similarly, if ν = 0 and there is a number N > 0 such that P (C_{T} > N) = 0, then the upper universal bound is non-attainable (use part (c) of Proposition 4.1).

5) Assume now that X (or Z, or S) is a process of finite variation, then

C = 0 and (|x| ∧ 1) ∗ ν_{T} < ∞ P -a.s.  (4.5)

Then we can define

B′ = X − ∑_{s≤·} ΔX_{s}.

It follows from the canonical representation of semimartingales that

B′ = B(ϕ) − ϕ(x) ∗ ν.  (4.6)

5.1) If X is a process of finite variation and there is a number δ > 0 such that P -a.s. ν([0, T] × (0, δ)) = 0 and P (B′_{T} > −δ) = 0, then the lower universal bound is non-attainable. Indeed, let Q ∈ P. Using the notation from subsection 3.2, we obtain from (3.4)–(3.7), (4.5) and (4.6) that Q-a.s.

|x| ∗ ν_{T}^{Q} < ∞  (4.7)

and

B′ + x ∗ ν^{Q} = 0.  (4.8)

Therefore,

x1_{{x≥δ}} ∗ ν_{T}^{Q} = |x|1_{{|x|<0}} ∗ ν_{T}^{Q} − B′_{T} ≥ δ Q-a.s.,

and the claim easily follows from (3.10) and the implication (i)⇒(iii) in Proposition 4.2. 5.2) Similarly, if X is a process of finite variation and there is a number δ > 0 such that P -a.s. ν([0, T] × (−δ, 0)) = 0 and P (B′_{T} < δ) = 0, then the lower universal bound is non-attainable.

5 Comparison lemma

Let T_{M} = {μ_{Q}: Q ∈ M}, μ_{Q} = L(Z_{T} | Q) as in Section 1. In subsection 4.2 we have considered a number of cases where the least upper (resp. the greatest lower) bound for T_{M} does not coincide with δ_{{0}} (resp. with δ_{{1}}). In such a situation the first step of the method proposed in subsection 1.4 for finding ℂ^{∗} (resp. ℂ_{∗}) is to show that μ_{Q} ≼ μ (resp. μ_{Q} ≽ μ) for all Q ∈ M, where μ ∈ S is a non-trivial candidate to be the least upper (resp. the greatest lower) bound for T_{M}. As we have already mentioned, a suitable description of T_{M} is usually not available if the market is not complete. The most essential information about Q in our disposal is contained in the relations (3.5) and (3.7) (at least if M = P). Thus it is desirable to have a comparison tool based on the corresponding local characteristics under Q. In general, this does not give us the possibility of comparing the measure μ_{Q} with a measure μ if no other specification of μ is given. However, fortunately, in many models the measures μ corresponding to the least upper and the greatest lower bounds can be described by the property μ = L(Z_{T}^{∗} | Q^{∗}), where Z^{∗} is either Z or the canonical process on a trajectory space and Q^{∗} is a certain probability measure (maybe on an extension of the original space); moreover, the process Z^{∗} under Q^{∗} has a rather special (e.g. Markovian) structure. In such a case there is an efficient comparison tool in terms of local characteristics presented in this section.

5.1. Let Z = {Z_{t}}_{0≤t≤T} and Z^{∗} = {Z_{t}^{∗}}_{0≤t≤T}, Z_{0} = Z_{0}^{∗} = 1, be two non-negative supermartingales defined on two stochastic basis B = (Ω, F_{T} , {F_{t}}_{0≤t≤T} , Q) and B^{∗} = (Ω^{∗}, F_{T}^{∗}, {F_{t}^{∗}}_{0≤t≤T} , Q^{∗}) respectively. We wish to compare the distributions μ = L(Z_{T} | Q) and μ^{∗} = L(Z_{T}^{∗} | Q^{∗}) in the sense of the partial ordering ≼ introduced in subsection 1.2. In the option pricing context Z is the process defined in Section 1 and Q ∈ M, thus μ = μ_{Q}. Since Q is an arbitrary equivalent martingale measure in general, there are no special assumptions on Z and Q. (In fact, we shall impose some restrictions, see (5.3) below, which exclude, for example, the case of discrete time. This is done for simplicity only. In Section 7 we shall consider an example with discrete time). The measure μ^{∗} plays the role of an upper or a lower bound for {μ_{Q}: Q ∈ M}. The assumptions on Q^{∗} and Z^{∗} are much more restrictive according to the introduction of this section. As it was explained in Sections 1 and 2, the least upper bound μ^{∗} for the family {μ_{Q}: Q ∈ M} need not satisfy the relations μ^{∗}({0}) = 0 and ∫ x μ^{∗}(dx) = 1. Thus it is reasonable not to assume that Z^{∗} is a Q^{∗}-martingale or that Z^{∗} is strictly positive if we look for conditions under which μ ≼ μ^{∗}. On the other hand, in the option pricing setting Q(Z_{T} > 0) = 1 and E_{Q}Z_{T} = 1, hence Q^{∗}(Z_{T}^{∗} > 0) = 1 and E_{Q^{∗}}Z_{T}^{∗} = 1 if μ^{∗} ≼ μ. Nevertheless, for completeness of the picture in our comparison lemma below both μ and μ^{∗} are allowed to have a positive mass at 0 and to have mean less than 1, i.e. Z and Z^{∗} are not necessarily martingales and may vanish. We shall assume that Z and Z^{∗} satisfy the stochastic differential equations

dZ_{t} = Z_{t−}dX_{t},  0 ≤ t ≤ T,  Z_{0} = 1,  (5.1)

and

dZ_{t}^{∗} = Z_{t−}^{∗}dX_{t}^{∗}, 0 ≤ t ≤ T, Z_{0}^{∗} = 1, (5.2)

i.e. Z = E(X) and Z^{∗} = E(X^{∗}), where X = {X_{t}}_{0≤t≤T} and X^{∗} = {X_{t}^{∗}}_{0≤t≤T} are local supermartingales on B and B^{∗} respectively, with characteristics (B(ϕ), C, ν) and (B^{∗}(ϕ), C^{∗}, ν^{∗}), and such that ΔX ≥ −1, and ΔX^{∗} ≥ −1. Let A = {A_{t}}_{0≤t≤T} and A^{∗} = {A_{t}^{∗}}_{0≤t≤T} be predictable non-increasing processes on B and B^{∗} respectively such that A_{0} = 0, A_{0}^{∗} = 0, X − A and X^{∗} − A^{∗} are local martingales on B and B^{∗}, see subsection 3.4 for the relationship between A and B(ϕ), A^{∗} and B^{∗}(ϕ). For simplicity we shall assume that

A_{t}(ω) = ∫_{0}^{t} a_{s}(ω) ds, C_{t}(ω) = ∫_{0}^{t} c_{s}(ω) ds, ν(ω, dt, dx) = K(ω, t, dx) dt, (5.3)

A_{t}^{∗}(ω^{∗}) = ∫_{0}^{t} a_{s}^{∗}(ω^{∗}) ds, C_{t}^{∗}(ω^{∗}) = ∫_{0}^{t} c_{s}^{∗}(ω^{∗}) ds, ν^{∗}(ω^{∗}, dt, dx) = K^{∗}(ω^{∗}, t, dx) dt,

(5.4)

where a = {a_{t}}_{0≤t≤T}, a^{∗} = {a_{t}^{∗}}_{0≤t≤T} are non-positive and c = {c_{t}}_{0≤t≤T}, c^{∗} = {c_{t}^{∗}}_{0≤t≤T} are nonnegative predictable processes on the corresponding spaces, and the transition kernels K = K(ω, t, dx) and K^{∗} = K^{∗}(ω, t, dx) satisfy the standard assumptions stated in Proposition II.2.9 in Jacod and Shiryaev (1987). We require an additional Markovian structure on Z^{∗}. Namely, we shall assume the existence of functions a^{∗} = a^{∗}(t, z) and c^{∗} = c^{∗}(t, z) and a kernel K^{∗} = K^{∗}(t, z, dx) such that

a_{t}^{∗}(ω^{∗}) = a^{∗}(t, Z_{t−}^{∗}(ω^{∗})),  c_{t}^{∗}(ω^{∗}) = c^{∗}(t, Z_{t−}^{∗}(ω^{∗})),  0 ≤ t ≤ T,  (5.5)

and

K^{∗}(ω^{∗}, t, dx) = K^{∗}(t, Z_{t−}^{∗}(ω^{∗}), dx),  0 ≤ t ≤ T.  (5.6)

This makes natural to assume, that the ∗-prices defined by

G(t, z) = E_{Q^{∗}}[f(Z_{T}^{∗}) | Z_{t}^{∗} = z],  (5.7)

where f belongs to a sufficiently large class C′ ⊆ C, satisfy a pricing equation (i.e. the backward Kolmogorov equation) of the form

G_{t}(t, z) + L_{t}G(t, z) = 0,  (5.8)

where

L_{t}G(t, z) = za^{∗}(t, z)G_{z}(t, z) + \frac{1}{2}z^{2}c^{∗}(t, z)G_{zz}(t, z) + ∫_{[−1,∞)}(ΛG)(t, z, x)K^{∗}(t, z, dx)

and

(ΛG)(t, z, x) = G(t, z(1+x))−G(t, z)−zxG_{z}(t, z), 0 ≤ t ≤ T, z > 0, x ≥ −1,

and G_{t}, G_{z} and G_{zz} are the corresponding partial derivatives. Actually, the formula (5.7) is not part of the assumptions; it serves only for the explanation how the function G appears and what are the boundary conditions for it. The last assumption is the propagation of convexity, stated in our framework as

G(t, z) is convex in z ∈ [0, ∞) for each t ∈ [0, T],  (5.9)

see El Karoui et al. (1998) and Martini (2000). Here we prefer not to discuss our assumptions. We shall check them later for some particular models.

Lemma 5.1 (Comparison) Assume that the following hypotheses are satisfied:

(A) Z is a nonnegative supermartingale on B satisfying (5.1), where X is a local supermartingale on B, ΔX ≥ −1, and the triplet (A, C, ν) satisfies (5.3).

(B) Z^{∗} is a nonnegative supermartingale on B^{∗} satisfying (5.2), where X^{∗} is a local supermartingale on B^{∗}, ΔX^{∗} ≥ −1, and the triplet (A^{∗}, C^{∗}, ν^{∗}) satisfies (5.4)–(5.6).

(C) For any function f ∈ C_{0}^{2} there is a real-valued function G(t, z), 0 ≤ t ≤ T, z ≥ 0, with the following properties:

(C1) G(T, z) = f(z), z ≥ 0; (C2) G(0, 1) = E_{Q^{∗}}f(Z_{T}^{∗}); (C3) G(t, 0) = f(0) for 0 < t ≤ T; (C4) G is bounded and continuous on [0, T] × [0, ∞), twice continuously differentiable in z and continuously differentiable in t on [0, T) × (0, ∞);

(C5) the pricing equation (5.8) holds true for 0 ≤ t < T, z > 0;

(C6) propagation of convexity (5.9) is satisfied.

(D) Comparison of predictable characteristics.

c_{t}(ω) ≤ c^{∗}(t, Z_{t−}(ω))  dt × dQ-a.e.  (5.10)

For any w ∈ (−1, 0), dt × dQ-a.e.

∫_{[−1,∞)}(w − x)^{+}K(ω, t, dx) ≤ ∫_{[−1,∞)}(w − x)^{+}K^{∗}(t, Z_{t−}(ω), dx). (5.11)

For any w ∈ (0, ∞), dt × dQ-a.e.

∫_{[−1,∞)}(x − w)^{+}K(ω, t, dx) − a_{t}(ω)

≤ ∫_{[−1,∞)}(x − w)^{+}K^{∗}(t, Z_{t−}(ω), dx) − a^{∗}(t, Z_{t−}(ω)).  (5.12)

Then L(Z_{T} | Q) ≼ L(Z_{T}^{∗} | Q^{∗}). If the opposite inequalities hold in (D) then L(Z_{T}^{∗} | Q^{∗}) ≼ L(Z_{T} | Q).

Remark 5.1 In the case when the processes Z and Z^{∗} are strictly positive martingales on [0, T] satisfying the hypothesis of the previous Lemma, denoting μ_{Q} = L(Z_{T} | Q) and μ_{Q^{∗}} = L(Z_{T} | Q^{∗}) we obtain J_{f}(μ_{Q}) = ∫ f(x) μ_{Q}(dx) and J_{f}(μ_{Q^{∗}}) = ∫ f(x) μ_{Q^{∗}}(dx). Then, the conclusion L(Z_{T} | Q) ≼ L(Z_{T}^{∗} | Q^{∗}) of the Lemma is equivalent to

E_{Q}f(Z_{T}) ≤ E_{Q^{∗}}f(Z_{T}^{∗}) for all f ∈ C.

Remark 5.2 It follows from the proof that, if the process Z is strictly positive then it is sufficient to assume that the function G is defined on the set [0, T] × (0, ∞), that it is bounded and continuous only on this set in (C4), and condition (C3) is not necessary. On the other hand, if Z and Z^{∗} are local martingales with respect to Q and Q^{∗} respectively, then a = a^{∗} = 0 and condition (5.12) takes a more simple form.

Proof. We shall prove only the first statement. The proof of the second one is completely similar. Let us fix a function f ∈ C_{0}^{2}. Due to Corollary 2.1 it is enough to check that

E_{Q}f(Z_{T}) ≤ E_{Q^{∗}}f(Z_{T}^{∗}).  (5.13)

For n = 1, 2, … define τ_{n} = inf {t: Z_{t} < 1/n} (inf ∅ = T), τ = lim_{n→∞} τ_{n}. It is well known that Q-a.s. {Z = 0} = [[τ, T]]. Moreover, in view of (5.1), Q-a.s. Z_{τ−} > 0 and [[0, τ]] = ⋃_{n}[[0, τ_{n}]]. Take also an increasing sequence {t_{n}}, t_{n} → T. Applying Itô’s formula to {G(t, Z_{t})} on B, we obtain for t ∈ [0, t_{n}]

G(t ∧ τ_{n}, Z_{t∧τ_{n}}) = G(0, 1) + ∫_{0}^{t∧τ_{n}} G_{t}(s, Z_{s−}) ds + ∫_{0}^{t∧τ_{n}} G_{z}(s, Z_{s−})Z_{s−}dX_{s}

+ \frac{1}{2} ∫_{0}^{t∧τ_{n}} G_{zz}(s, Z_{s−})Z_{s−}^{2}c_{s} ds + ∫_{0}^{t∧τ_{n}} ∫_{[−1,∞)}(ΛG)(s, Z_{s−}, x)μ(ds, dx), (5.14)

where μ is the jump measure of X. Due to the properties of G, all the terms in (5.14) excluding the last one are special semimartingales in t. Hence, the last term is also a special semimartingale with the compensator

∫_{0}^{t∧τ_{n}} ∫_{[−1,∞)}(ΛG)(s, Z_{s−}, x)K(s, dx) ds,

and we may rewrite (5.14) in the form

G(t ∧ τ_{n}, Z_{t∧τ_{n}}) = G(0, 1) + ∫_{0}^{t∧τ_{n}} G_{t}(s, Z_{s−}) ds + ∫_{0}^{t∧τ_{n}} G_{z}(s, Z_{s−})Z_{s−}a_{s}ds

+\frac{1}{2} ∫_{0}^{t∧τ_{n}} G_{zz}(s, Z_{s−})Z_{s−}^{2}c_{s} ds + ∫_{0}^{t∧τ_{n}} ∫_{[−1,∞)}(ΛG)(s, Z_{s−}, x)K(s, dx) ds + m_{t}^{n},

where m^{n} is a local martingale on B. Now replace G_{t}(s, Z_{s−}) according to (5.8), to obtain

G(t ∧ τ_{n}, Z_{t∧τ_{n}}) = G(0, 1) + \frac{1}{2} ∫_{0}^{t∧τ_{n}} G_{zz}(s, Z_{s−})Z_{s−}^{2}(c_{s} − c^{∗}(s, Z_{s−})) ds +

+ ∫_{0}^{t∧τ_{n}} [ ∫_{[−1,∞)}(ΛG)(s, Z_{s−}, x)K(s, dx) + a_{s}G_{z}(s, Z_{s−})Z_{s−}

−{ ∫_{[−1,∞)}(ΛG)(s, Z_{s−}, x)K^{∗}(s, Z_{s−}, dx)) + a^{∗}(s, Z_{s−})G_{z}(s, Z_{s−})Z_{s−}}] ds + m_{t}^{n}

= G(0, 1) + D_{t}^{n,1} + D_{t}^{n,2} + m_{t}^{n} = G(0, 1) + D_{t}^{n} + m_{t}^{n}.

Assume for the moment that the process D^{n} = {D_{t}^{n}}_{0≤t≤T} is decreasing Q-a.s. Then {G(t∧τ_{n}, Z_{t∧τ_{n}})}_{0≤t≤t_{n}} is a Q-local supermartingale, hence a Q-supermartingale since it is bounded. Observe also, that as n → ∞,

G(t_{n} ∧ τ_{n}, Z_{t_{n}∧τ_{n}}) → G(T, Z_{T})1_{{τ=T }} + G(τ, 0)1_{{τ<T }} = f(Z_{T}),

because the process Z is stochastically continuous, and boundary conditions in (C). Then, as G is bounded,

E_{Q}f(Z_{T}) = lim_{n} E_{Q}G(t_{n} ∧ τ_{n}, Z_{t_{n}∧τ_{n}}) ≤ G(0, 1) = E_{Q^{∗}}f(Z_{T}^{∗}),

that yields (5.13). It remains to show that D^{n} is decreasing Q-a.s. In view of (5.9) and (5.10) it is clear that D^{n,1} = {D_{t}^{n,1}}_{0≤t≤T} is decreasing Q-a.s. Let Γ be the set of all (ω, t) ∈ Ω × [0, T] for which (5.11) and (5.12) hold simultaneously for all rational w; we have λ × Q((Ω × [0, T]) \ Γ) = 0, where λ is the Lebesgue measure on [0, T]. Denote by Ĉ the set of all nonnegative continuous convex functions g on [−1, ∞) such that g(0) = 0 and \frac{g(∞)}{∞} := lim_{x↑∞} \frac{g(x)}{x} < ∞. Then for all (ω, t) ∈ Γ

∫_{[−1,∞)} g(x)K(ω, t, dx) − \frac{g(∞)}{∞}a_{t}(ω)

≤ ∫_{[−1,∞)} g(x)K^{∗}(t, Z_{t−}(ω), dx) − \frac{g(∞)}{∞}a^{∗}(t, Z_{t−}(ω)).  (5.15)

Indeed, let Ĉ_{n} be the subset of Ĉ consisting of piecewise linear functions with break points in the set {k/n: k = ±1, ±2, …, ±(n − 1), n, n + 1, …, n^{2}}. In view of (5.11) and (5.12), (5.15) is valid for any g ∈ ⋃_{n} Ĉ_{n}. On the other hand, it is clear that for any function g ∈ Ĉ one can find a sequence {g_{n}}_{n≥1} such that g_{n} ∈ Ĉ_{n}, g_{n}(x) ≤ g_{n+1}(x), n ≥ 1, lim_{n→∞} g_{n}(x) = g(x) for all x ∈ [−1, ∞), and \frac{g(∞)}{∞} = lim_{n→∞} \frac{g_{n}(∞)}{∞}.

Since G(t, z) is bounded, continuous and convex in z, the function x ⇝ (ΛG)(t, z, x) belongs to Ĉ for any t and z. In particular, x ⇝ (ΛG)(t, Z_{t−}(ω), x) belongs to Ĉ for any t and ω and \frac{ΛG(t,z,∞)}{∞} = −zG_{z}(t, z). So, (5.15) gives

∫_{[−1,∞)}(ΛG)(t, Z_{t−}(ω), x)K(ω, t, dx) + Z_{t−}(ω)G_{z}(t, Z_{t−}(ω))a_{t}(ω)

≤ ∫_{[−1,∞)}(ΛG)(t, Z_{t−}(ω), x)K^{∗}(t, Z_{t−}(ω), dx) + Z_{t−}(ω)G_{z}(t, Z_{t−}(ω))a^{∗}(t, Z_{t−}(ω))

for all (t, ω) ∈ Γ. It follows that D^{n,2} is decreasing Q-a.s. The lemma has been proved.

Remark 5.3 It follows from the proof that in condition (C) one can replace the class C_{0}^{2} by any subclass C′ ⊂ C_{0}^{2} with the property that the inequality ∫ f(x) μ_{𝔼}(dx) ≥ ∫ f(x) μ𝔼̃(dx) for any f ∈ C′ always implies 𝔼 ≽ 𝔼̃.

Remark 5.4 Of course, the inequalities (5.10)–(5.12) in (D) are only sufficient for L(Z_{T} | Q) ≼ L(Z_{T}^{∗} | Q^{∗}). We prove, in fact, much more. Informally speaking, for any t ∈ [0, T] we construct a new process Z^{t} which coincides with Z on [0, t] and evolves according to (5.2), (5.4)–(5.6) on [t, T] starting from Z_{t} at the moment t. The proof shows that the family L(Z_{T}^{t}) is monotone in t with respect to the partial ordering ≼.

Remark 5.5 To our knowledge, this type of arguments was introduced in Bellamy and Jeanblanc (2000), where Z^{∗} is a diffusion and Z is a diffusion with jumps, and used afterwards in Jakubenas (1998) in the context of Lévy processes, see also El Karoui et al. (1998).

5.2. We want now to examine some examples where the hypotheses (B) and (C) on Z^{∗} in Comparison Lemma hold. For notational simplicity we drop the ∗ in this subsection.

5.2.1. Processes with independent increments

Assume that X is a process with independent increments (from now on PII), with deterministic characteristics (B(ϕ), C, ν), ν is concentrated on [0, T] × [−1, ∞) and the process A defined by (3.13) is non-increasing to ensure that Z = E(X) is a nonnegative supermartingale. We suppose that X is locally homogeneous in the sense of § 3.4 in Skorokhod (1991). This amounts to say that there exist a nonpositive function a = a(t), a nonnegative function c = c(t) and a transition kernel K = K(t, dx) such that

A_{t} = ∫_{0}^{t} a(s) ds,  C_{t} = ∫_{0}^{t} c(s) ds,  ν(dt, dx) = K(t, dx) dt.

Given t ∈ [0, T] denote by X̃ = {X̃_{u}}_{0≤u≤T} the PII given by

X̃_{u} = \{ \begin{matrix}0, & if & 0 ≤ u ≤ t, \\ X_{u} − X_{t}, & if & t < u ≤ T . \end{matrix}  (5.16)

In accordance with (5.7) define

G(t, z) = E(f(Z_{T}) | Z_{t} = z) = E[f(zE(X̃)_{T})], f ∈ C_{0}^{2}. (5.17)

The most part of the properties of G required in (C) including convexity in z are immediate from (5.17); only the pricing equation (5.8) needs to be justified. First assume that Z is strictly positive. Then put \overline{X}_{t} = log Z_{t}. The process \overline{X} = \{\overline{X}_{t}\}_{0≤t≤T} is a PII with the characteristics given by (3.3). Since G(t, z) = Ef (ze^{\overline{X}_{T} −\overline{X}_{t}}) = Eh(log z + \overline{X}_{T} − \overline{X}_{t}), where h(x) = f(e^{x}), x ∈ ℝ, one can apply Theorem 25 in Skorokhod (1991, p. 161) to get the equation (3.55) ibid-em. After the corresponding change of variables and using (3.3) and (3.13), one obtains

G_{t}(t, z) + za(t)G_{z}(t, z) + \frac{1}{2}z^{2}c(t)G_{zz}(t, z) + ∫_{[−1,∞)}(ΛG)(t, z, x)K(t, dx) = 0. (5.18)

In the general case, when Z can vanish, let us take a truncation function ϕ(x) with ϕ(−1) = 0 and define the processes X^{(1)} = {X_{t}^{(1)}}_{0≤t≤T} and X^{(2)} = {X_{t}^{(2)}}_{0≤t≤T} as independent PII with characteristics (B(ϕ), C, 1_{[0,T]×(−1,∞)}ν) and (0, 0, 1_{[0,T]×{−1}}ν) respectively defined on some probability space. It is clear that X =^{d} X^{(1)} + X^{(2)};

moreover, X̃ =^{d} X̃^{(1)} + X̃^{(2)} and E(X̃) =^{d} E(X̃^{(1)} + X̃^{(2)}) = E(X̃^{(1)})E(X̃^{(2)}) defining

X̃^{(i)} as in (5.16) for i = 1, 2. The process E(X̃^{(2)}) takes only values 1 and 0, and P (E(X̃^{(2)})_{T} = 1) = exp (− ∫_{t}^{T} K(s, {−1}) ds). Therefore,

G(t, z) = G^{(1)}(t, z) exp (− ∫_{t}^{T} K(s, {−1}) ds)
+ f(0) {1 − exp (− ∫_{t}^{T} K(s, {−1}) ds)} ,

where G^{(1)}(t, z) = Ef(zE(X̃^{(1)})_{T}). Since X^{(1)} is strictly positive, we already know the pricing equation for G^{(1)}: replace a(t) by a(t)−K(t, {−1}) and ∫_{[−1,∞)} by ∫_{(−1,∞)} in (5.18). Now the pricing equation (5.18) for G follows from the previous formula.

5.2.2. Diffusions

Assume now for Z a dynamics given by

dZ_{t} = Z_{t}σ(t, Z_{t})dW_{t}, 0 ≤ t ≤ T, Z_{0} = 1,

where W is a standard Wiener process on a stochastic basis B. Assume that the function σ(t, z) is continuous in (t, z), and bounded from above, i.e.

σ(t, z) ≤ σ_{1} < ∞.

Assume that the function \frac{∂}{∂z}(zσ(t, z)) is continuous for all (t, z), and Lipschitz continuous and bounded in z ∈ (0, ∞) uniformly in t ∈ [0, T]. Under these conditions, propagation of convexity (5.9) holds (see El Karoui et al. (1998)). We give a proof of this fact for the case of time-homogeneous diffusions, i.e. when σ does not depend on t. Assume then that

dZ_{t} = Z_{t}σ(Z_{t})dW_{t}, 0 ≤ t ≤ T, Z_{0} = 1,  (5.19)

with σ = σ(z) is continuous, and the function zσ(z) is Lipschitz (or satisfies a local Lipschitz condition). Under this hypothesis, there exists a unique strong solution to (5.19), see Theorem IV.3.1 in Ikeda and Watanabe (1981). Let P (x, t, Γ) be the corresponding transition probability function. Define G according to (5.7):

G(t, z) = ∫ f(x)P (z, T − t, dx), f ∈ C_{0}^{2}.

Then the backward Kolmogorov equation is

G_{t}(t, z) + \frac{1}{2}z^{2}σ^{2}(z)G_{zz}(t, z) = 0,  (5.20)

see e.g. Shiryaev (1999, Chapter III, § 3f). According to (5.20), in order to check the convexity of G in z for a fixed t, it is sufficient to show that G is decreasing in t for each fixed z. Consider 0 ≤ t ≤ t + h ≤ T. Using the Kolmogorov–Chapman equation, Jensen’s inequality and the martingale property of Z, we get

G(t, z) = ∫ f(x)P (z, T − t, dx) = ∫ ∫ f(x)P (y, h, dx)P (z, T − t − h, dy)
≥ ∫ f (∫ xP (y, h, dx)) P (z, T − t − h, dy) = ∫ f(y)P (z, T − t − h, dy)
= G(t + h, z).

5.3. In this subsection we return to the initial setting of Section 1. We then assume that Z = {Z_{t}}_{0≤t≤T}, Z_{0} = 1, is a strictly positive semimartingale on a stochastic basis B = (Ω, F_{T} , {F_{t}}_{0≤t≤T} , P), P ≠ ∅ is the set of all equivalent martingale measures for Z. The process Z^{∗} = {Z_{t}^{∗}}_{0≤t≤T}, Z_{0}^{∗} = 1, defined on a stochastic basis B^{∗} = (Ω^{∗}, F_{T}^{∗}, {F_{t}^{∗}}_{0≤t≤T} , Q^{∗}) is the same as in subsection 5.1, in particular, Z^{∗} is a nonnegative supermartingale satisfying (5.2), and (5.4)–(5.6) hold. Here (B(ϕ), C, ν) is the triplet of local characteristics of X, where Z = E(X), relative to the measure P, while the triplet of X with respect to Q ∈ P is denoted by (B^{Q}(ϕ), C^{Q}, ν^{Q}). We assume that

C_{t}(ω) = ∫_{0}^{t} c_{s}(ω) ds,  ν(ω, dt, dx) = K(ω, t, dx) dt,  (5.21)

where c = {c_{t}}_{0≤t≤T} is a predictable nonnegative process and K = K(ω, t, dx) is a transition kernel satisfying the standard assumptions. Then there is a predictable process b = {b_{t}}_{0≤t≤T} such that

B_{t}(ϕ)(ω) = ∫_{0}^{t} b_{s}(ω) ds  (5.22)

(otherwise P = ∅ as can be easily seen from (3.5) and (3.7)). Using the notation as in (3.4) and (3.5), we see that for any Q ∈ P

C_{t}^{Q}(ω) = ∫_{0}^{t} c_{s}^{Q}(ω) ds,  ν^{Q}(ω, dt, dx) = K^{Q}(ω, t, dx) dt,  P -a.s., (5.23)

where

c_{t}^{Q}(ω) = c_{t}(ω),  K^{Q}(ω, t, dx) = Y^{Q}(ω, t, x)K(ω, t, dx).  (5.24)

Note also that X is a Q-local martingale for any Q ∈ P, hence the corresponding process A^{Q} is equal to 0.

Our goal here is to describe some particular cases where the inequalities (5.10)–(5.12) (or the opposite inequalities) with c, K, a replaced by c^{Q}, K^{Q}, 0 respectively hold for any Q ∈ P. More precisely, we are interested in the situation when, for every Q ∈ P,

c_{t}^{Q}(ω) ≤ c^{∗}(t, Z_{t−}(ω)) dt × dQ-a.e.,  (5.25)

∫_{[−1,∞)}(w − x)^{+}K^{Q}(ω, t, dx) ≤ ∫_{[−1,∞)}(w − x)^{+}K^{∗}(t, Z_{t−}(ω), dx) dt × dQ-a.e.

(5.26)

for any w ∈ (−1, 0),

∫_{[−1,∞)}(x − w)^{+}K(ω, t, dx)
≤ ∫_{[−1,∞)}(x − w)^{+}K^{∗}(t, Z_{t−}(ω), dx) − a^{∗}(t, Z_{t−}(ω)) dt × dQ-a.e. (5.27)

for any w ∈ (0, ∞), or

c_{t}^{Q}(ω) ≥ c^{∗}(t, Z_{t−}(ω)) dt × dQ-a.e.,  (5.28)

∫_{[−1,∞)}(w − x)^{+}K^{Q}(ω, t, dx) ≥ ∫_{[−1,∞)}(w − x)^{+}K^{∗}(t, Z_{t−}(ω), dx) dt × dQ-a.e.

(5.29)

for any w ∈ (−1, 0),

∫_{[−1,∞)}(x − w)^{+}K(ω, t, dx)
≥ ∫_{[−1,∞)}(x − w)^{+}K^{∗}(t, Z_{t−}(ω), dx) − a^{∗}(t, Z_{t−}(ω)) dt × dQ-a.e. (5.30)

for any w ∈ (0, ∞). The attainability of the corresponding bounds is not discussed. The first two cases are easy consequences of (5.24).

1) Let X^{∗} be a continuous Q^{∗}-local martingale (i.e. ν^{∗} = 0 and A^{∗} = 0). If

c_{t}(ω) ≥ c^{∗}(t, Z_{t−}(ω)) dt × dP -a.e.,

then we have (5.28)–(5.30) for every Q ∈ P.

2) Let X be a continuous process (i.e. ν = 0). If

c_{t}(ω) ≤ c^{∗}(t, Z_{t−}(ω)) dt × dP -a.e.,

then we have (5.25)–(5.27) for every Q ∈ P. Assume now that X is a process of finite variation. Then (cf. subsection 4.2)

c = 0 and ∫ (|x| ∧ 1)K(ω, t, dx) < ∞ dt × dP -a.e.

Define B′ = X − ∑_{s≤·} ΔX_{s}. From (4.6), we obtain

B′_{t}(ω) = ∫_{0}^{t} b′_{s}(ω)ds, b′_{t}(ω) = b_{t}(ω) − ∫ ϕ(x)K(ω, t, dx).

Moreover, for any Q ∈ P we have (4.7) and (4.8), hence

∫ |x|K^{Q}(ω, t, dx) < ∞ and ∫ xK^{Q}(ω, t, dx) = −b′_{t}(ω) dt × dP -a.e.

The last equality is crucial for the remaining cases.

3) Let X be a process of finite variation and P -a.s. ΔX ≥ 0. If

−b′_{t}(ω) ≤ a^{∗}(t, Z_{t−}(ω)) dt × dP -a.e.,

then we have (5.25)–(5.27) for every Q ∈ P. Indeed, (5.25) and (5.26) are trivial and, for w > 0,

∫_{[−1,∞)}(x − w)^{+}K^{Q}(ω, t, dx) = ∫_{(0,∞)}(x − w)^{+}K^{Q}(ω, t, dx)
≤ ∫_{(0,∞)} xK^{Q}(ω, t, dx) = −b′_{t}(ω) ≤ a^{∗}(t, Z_{t−}(ω))

dt × dP -a.e. and dt × dQ-a.e.

4) Let X be a process of finite variation and P -a.s. 0 ≤ ΔX ≤ L for some L < ∞. If

−b′_{t}(ω) ≤ LK^{∗}(t, Z_{t−}(ω), {L}) dt × dP -a.e.,

then we have (5.25)–(5.27) for every Q ∈ P. Indeed, only (5.27) for w ∈ (0, L) has to be checked. Since (x − w)^{+} ≤ L^{−1}(L − w)x, x ∈ [0, L], we obtain

∫_{[−1,∞)}(x − w)^{+}K^{Q}(ω, t, dx) = ∫_{(0,L]}(x − w)^{+}K^{Q}(ω, t, dx)

≤ \frac{L − w}{L} ∫_{(0,L]} xK^{Q}(ω, t, dx) = −\frac{L − w}{L}b′_{t}(ω) ≤ (L − w)K^{∗}(t, Z_{t−}(ω), \{L\})

≤ ∫_{[−1,∞)}(x − w)^{+}K^{∗}(t, Z_{t−}(ω), dx) dt × dQ-a.e.

5) Let X be a process of finite variation and P -a.s. −L ≤ ΔX ≤ 0 for some L ∈ (0, 1]. If

b′_{t}(ω) ≤ LK^{∗}(t, Z_{t−}(ω), {−L}) dt × dP -a.e.,

then we have (5.25)–(5.27) for every Q ∈ P. The proof is similar to the previous case.

6) Assume that X is a process of finite variation, there is a number δ > 0 such that P {∃t: ΔX_{t} ∈ (0, δ)} = 0, and X^{∗} is a local martingale of finite variation with jumps of a fixed size δ. If

−b′_{t}(ω) ≥ δK^{∗}(t, Z_{t−}(ω), {δ}) dt × dP -a.e.,

then we have (5.28)–(5.30) for every Q ∈ P. Indeed, (5.28) and (5.29) are trivial, and, for w ∈ (0, δ),

∫_{[−1,∞)}(x − w)^{+}K^{Q}(ω, t, dx) = ∫_{[δ,∞)}(x − w)^{+}K^{Q}(ω, t, dx)

≥ ∫_{[δ,∞)}(x − w)K^{Q}(ω, t, dx) ≥ (1 − \frac{w}{δ}) ∫_{[δ,∞)} xK^{Q}(ω, t, dx)

≥ (1 − \frac{w}{δ}) ∫_{[−1,∞)} xK^{Q}(ω, t, dx) = −(1 − \frac{w}{δ})b′_{t}(ω)

≥ (δ − w)K^{∗}(t, Z_{t−}(ω), {δ}) = ∫_{[−1,∞)}(x − w)^{+}K^{∗}(t, Z_{t−}(ω), dx)

dt × dQ-a.e.

7) Assume that X is a process of finite variation, there is a number δ ∈ (0, 1) such that P {∃t: ΔX_{t} ∈ (−δ, 0)} = 0, and X^{∗} is a local martingale of finite variation with jumps of a fixed size −δ. If

b′_{t}(ω) ≥ δK^{∗}(t, Z_{t−}(ω), {−δ}) dt × dP -a.e.,

then we have (5.28)–(5.30) for every Q ∈ P. The proof is similar to the previous case.

5.4. In this subsection we show that the conditions (5.10)–(5.12) in the Comparison Lemma are only sufficient even if the local characteristic of X and X^{∗} are homogeneous in time and, moreover, deterministic, i.e. if X and X^{∗} are Lévy processes.

Assume then that X = {X_{t}}_{0≤t<∞} is a Lévy process on a stochastic basis B = (Ω, F, {F_{t}}_{0≤t<∞}, P); moreover, X is a supermartingale and ΔX ≥ −1. Then its characteristics (B(ϕ), C, ν) can be chosen in the form

B_{t}(ϕ) = b(ϕ)t,  C_{t} = ct,  ν(dt, dx) = dt F (dx),  (5.31)

where b(ϕ) ∈ ℝ, c ∈ ℝ_{+}, F is a σ-finite measure on [−1, ∞) \ {0}, which integrates x^{2} ∧ 1, and a := −b(ϕ) − ∫ (x − ϕ(x))F (dx) ≥ 0, cf. (3.13). Conversely, if the triplet (b(ϕ), c, F) satisfies these conditions, then there is a Lévy process X with the characteristics as in (5.31), which is a supermartingale with ΔX ≥ 1. Let Z = E(X) and define a statistical experiment 𝔼_{t}, t ≥ 0, by μ_{𝔼_{t}} = μ_{t} := L(Z_{t} | P). The statistical experiment 𝔼_{t} is infinitely divisible (see e.g. Janssen et al. (1985)). It is not difficult to show that if 𝔼 is an infinitely divisible binary experiment and 𝔼 is not totally informative, then one can find the triplet (b(ϕ), c, F) with the above properties such that 𝔼 = 𝔼_{1}. Similarly, we assume that X^{∗} = {X_{t}^{∗}}_{0≤t<∞} is a Lévy process and a supermartingale with ΔX^{∗} ≥ −1 (without loss of generality, we may suppose that X^{∗} is given on the same stochastic basis). The corresponding triplet is denoted by (b^{∗}(ϕ), c^{∗}, F^{∗}), a^{∗} := b^{∗}(ϕ) + ∫ (x − ϕ(x))F^{∗}(dx). Put Z^{∗} = E(X^{∗}) and define μ_{t}^{∗} = L(Z_{t}^{∗} | P). It follows from Comparison Lemma and considerations in subsection 5.2 that μ_{t} ≼ μ_{t}^{∗} for every t > 0 if

c ≤ c^{∗},  (5.32)
∫_{[−1,∞)}(w − x)^{+}F (dx) ≤ ∫_{[−1,∞)}(w − x)^{+}F^{∗}(dx) for any w ∈ (−1, 0), (5.33)

∫_{[−1,∞)}(x − w)^{+}F (dx) − a ≤ ∫_{[−1,∞)}(x − w)^{+}F^{∗}(dx) − a^{∗} for any w ∈ (0, ∞).

(5.34)

We now prove that the reciprocal of this last statement is true in the context of Lévy processes. Assume μ_{t} ≼ μ_{t}^{∗} for all t > 0 (or for a sequence {t_{n}} with t_{n} → 0). Then (5.32)–(5.34) are satisfied. Indeed, let f ∈ C_{0}^{2}. Using arguments based on Itô’s formula, or the ones that we used deducing the pricing equation, one can show that

lim_{t↓0} \frac{Ef(Z_{t}) − 1}{t} = af′(1) + \frac{1}{2}cf′′(1) + ∫_{[−1,∞)}(f(1 + x) − f(1) − f′(1)x) F (dx).

A similar formula holds for Z^{∗}. Therefore, if μ_{t_{n}} ≼ μ_{t_{n}}^{∗} for a sequence {t_{n}} converging to 0 then

af′(1) + \frac{1}{2}cf′′(1) + ∫_{[−1,∞)}(f(1 + x) − f(1) − f′(1)x) F (dx) ≤

≤ a^{∗}f′(1) + \frac{1}{2}c^{∗}f′′(1) + ∫_{[−1,∞)}(f(1 + x) − f(1) − f′(1)x) F^{∗}(dx) (5.35)

for any f ∈ C_{0}^{2}. Smoothing f(x) = (1 + w − x)^{+} near the break point, we come to the inequalities (5.33) and (5.34). Applying (5.35) for

f_{n}(x) = \{ \begin{matrix}\frac{2}{n}(1 − x), & if & 0 ≤ x < 1 − \frac{π}{2n}, \\ \frac{1}{n}(1 − x + \frac{π}{2n}) − \frac{1}{n^{2}} cos n(x − 1), & if & 1 − \frac{π}{2n} ≤ x ≤ 1 + \frac{π}{2n}, \\ 0, & if & x > 1 + \frac{π}{2n}, \end{matrix}

and passing to the limit as n → ∞, we get (5.32). However, it may happen that μ_{t} ≼ μ_{t}^{∗} for a fixed t > 0 but the conditions (5.32)– (5.34) are not satisfied. Let us take a truncation function ϕ such that ϕ(x) = x on [−1, e − 1] and put b(ϕ) = 0, c = σ^{2}, F = 0, b^{∗}(ϕ) = c^{∗} = 0, F^{∗} = δ_{{e^{−1}−1}} + δ_{{e−1}}, where σ > 0 is not fixed yet. Then a = a^{∗} = 0, and for any σ > 0 inequality (5.32) does not hold. To emphasize the dependence of the first model on σ, we write μ_{t}^{σ} and Z_{t}^{σ} instead of μ_{t} and Z_{t}. Let us fix t, say, t = 1. We assert that μ_{1}^{σ} ≼ μ_{1}^{∗} if σ is small enough. To prove it, we first note, cf. (3.2) and (3.3), that

Z_{t}^{σ} = exp (σW_{t} − \frac{σ^{2}}{2} t) and Z_{t}^{∗} = exp (π_{t}^{(1)} − π_{t}^{(2)} − γt), γ = e + e^{−1} − 2,

where W is a standard Wiener process, π^{(1)} and π^{(2)} are independent Poisson processes with intensity 1. Let G_{σ}(x) = P (Z_{1}^{σ} ≤ x) and G^{∗}(x) = P (Z_{1}^{∗} ≤ x). We have

G_{1}(x) = P (W_{1} ≤ log x + 1/2) = P (W_{1} ≥ − log x − 1/2),
1 − G_{1}(x) = P (W_{1} > log x + 1/2),
G^{∗}(x) = P (π_{1}^{(1)} − π_{1}^{(2)} ≤ log x + γ) ≥ P (π_{1}^{(1)} = 0)P (π_{1}^{(2)} ≥ − log x − γ)
= e^{−1}P (π_{1}^{(2)} ≥ − log x − γ),

1 − G^{∗}(x) = P (π_{1}^{(1)} − π_{1}^{(2)} > log x + γ) ≥ e^{−1}P (π_{1}^{(1)} > log x + γ).

Since Gaussian tails decrease faster than Poisson tails,

lim_{x↓0} \frac{G_{1}(x)}{G^{∗}(x)} = lim_{x↑∞} \frac{1 − G_{1}(x)}{1 − G^{∗}(x)} = 0.

Therefore,

lim_{y↓0} \frac{∫_{0}^{y} G_{1}(x) dx}{∫_{0}^{y} G^{∗}(x) dx} = lim_{y↑∞} \frac{∫_{y}^{∞}[1 − G_{1}(x)] dx}{∫_{y}^{∞}[1 − G^{∗}(x)] dx} = 0,  (5.36)

since

∫_{0}^{∞}[1 − G_{1}(x)] dx = ∫_{0}^{∞}[1 − G^{∗}(x)] dx = 1.  (5.37)

In view of (5.36) and (5.37), there are numbers 0 < δ < L < ∞ such that

∫_{0}^{y} G_{1}(x) dx ≤ ∫_{0}^{y} G^{∗}(x) dx for any y ∈ [0, δ) ∪ (L, ∞).  (5.38)

On the other hand, it is easy to see that μ_{1}^{σ} ≼ μ_{1}^{σ′} if σ ≤ σ′ and μ_{1}^{σ} ⇒ δ_{{1}} as

σ ↓ 0. Hence lim_{σ↓0} G_{σ}(x) = \{ \begin{matrix}0, & x < 1, \\ 1, & x > 1, \end{matrix} and ∫_{0}^{y} G_{σ}(x) dx converges to (y − 1)^{+}

uniformly in y on any compact interval as σ ↓ 0. But ∫_{0}^{y} G^{∗}(x) dx ≥ (y − 1)^{+} for any y > 0 by Proposition 2.2 and it is easy to see that the equality is not possible. Therefore, there is σ_{0} ≤ 1 such that

∫_{0}^{y} G_{σ}(x) dx ≤ ∫_{0}^{y} G^{∗}(x) dx for any y ∈ [δ, L]

if 0 < σ ≤ σ_{0}. In view of Proposition 2.2 and (5.38) we have

∫_{0}^{y} G_{σ}(x) dx ≤ ∫_{0}^{y} G_{1}(x) dx ≤ ∫_{0}^{y} G^{∗}(x) dx for any y ∈ [0, δ) ∪ (L, ∞)

if σ ≤ 1. Combining two last inequalities and using Proposition 2.2, we obtain μ_{1}^{σ} ≼ μ_{1}^{∗} if 0 < σ ≤ σ_{0}.

6 Attainability of non-trivial bounds

In Section 5 we have considered a certain machinery which can be useful for realizing the first step of our method in concrete models. Roughly speaking, the corresponding distributions are comparable if the triplets of local characteristics are comparable in a certain sense. The second step of the method proposed consists in finding a sequence of measures Q^{n} in M such that μ_{Q^{n}} ⇒ μ^{∗}, where μ^{∗} is a candidate to be the greatest lower or the least upper bound for the family {μ_{Q}: Q ∈ M}. Of course, it is natural to expect this convergence if we have a convergence of the corresponding triplets in the right sense. The aim of this section is to adapt some of known theorems on the weak convergence of likelihood processes, see e.g. Jacod and Shiryaev (1987), Jacod (1989), Coquet and Jacod (1990), Kramkov (1993), to the option pricing context in accordance with this point of view. Throughout the section we assume that Z = E(X), X_{0} = 0, is a strictly positive semimartingale on a stochastic basis (Ω, F_{T} , {F_{t}}_{0≤t≤T} , P), {Q^{n}}_{n≥1} is a sequence of equivalent probability measures on (Ω, F_{T}), i.e. Q^{n} ∼ P and Z is a Q^{n}-martingale for every n ≥ 1. The triplets of X with respect to Q^{n} are denoted by (B^{n}(ϕ), C, ν^{n}) (the second characteristics do not depend on n in view of (3.5)); they satisfy (3.6) and (3.7). The truncation function ϕ is assumed to be continuous. The limiting measure μ^{∗} is described in the spirit of the previous section by μ^{∗} = L(Z_{T}^{∗} | Q^{∗}), where Z^{∗} is a nonnegative supermartingale on a stochastic basis B^{∗} = (Ω^{∗}, F_{T}^{∗}, {F_{t}^{∗}}_{0≤t≤T} , Q^{∗}), representable as Z^{∗} = E(X^{∗}), where X^{∗} is a local supermartingale with ΔX^{∗} ≥ −1. We shall consider separately two cases: X^{∗} is a PII and Z^{∗} is a diffusion.

6.1. We assume here that X^{∗} is a stochastically continuous process with independent increments with a deterministic triplet (B^{∗}(ϕ), C^{∗}, ν^{∗}); A^{∗} := −B^{∗}(ϕ)−(x−ϕ(x))∗ν^{∗}.

Proposition 6.1 Under the above assumptions, let

sup_{t≤T} ∫_{{|x|>ε}} |x|ν^{n}({t} × dx) ⟶^{Q^{n}} 0  for all ε > 0,  (6.1)
C_{T} + ϕ^{2}(x) ∗ ν_{T}^{n} ⟶^{Q^{n}} C_{T}^{∗} + ϕ^{2}(x) ∗ ν_{T}^{∗} ,  (6.2)
(x − ϕ(x)) ∗ ν_{T}^{n} ⟶^{Q^{n}} (x − ϕ(x)) ∗ ν_{T}^{∗} + A_{T}^{∗} ,  (6.3)
g(x) ∗ ν_{T}^{n} ⟶^{Q^{n}} g(x) ∗ ν_{T}^{∗}  (6.4)

for any continuous bounded function g: [−1, ∞) → ℝ vanishing in a neighborhood of 0. Then L(Z_{T} | Q^{n}) ⇒ L(Z_{T}^{∗} | Q^{∗}).

Remark 6.1 Under (6.4), conditions (6.2) and (6.3) do not depend on the choice of a continuous truncation function ϕ.

Remark 6.2 If Z^{∗} is a strictly positive martingale, this proposition is an immediate consequence of Theorem X.2.12 in Jacod and Shiryaev (1987). The method of the proof of that theorem (to check the weak convergence of the terminal value of log Z under Q^{n} to the terminal value of the PII log Z^{∗} under Q^{∗}) can be applied also in the case where Z^{∗} is a strictly positive supermartingale. The limiting process Z^{∗} under our hypotheses was considered in Coquet and Jacod (1990, Theorem 3.6 and Lemma 3.11), where only the functional convergence was proved under more restrictive assumptions.

Proof. We give only a sketch of the proof. It is easy to check that under (6.2) and (6.3), (6.4) implies g(x) ∗ ν_{T}^{n} ⟶^{Q^{n}} g(x) ∗ ν_{T}^{∗} for any continuous bounded function g: [−1, ∞) → ℝ satisfying g(x) = o(x^{2}) as x → 0 and g(x) = o(x) as x → +∞. Hence,

h_{T}^{n}(α) ⟶^{Q^{n}} h_{T}^{∗} (α),  (6.5)

where α ∈ (0, 1), h^{n}(α) = \frac{α(1−α)}{2}C + \{α + (1 − α)(1 + x) − (1 + x)^{1−α}\} ∗ ν^{n} is the Hellinger process of order α for Q^{n} and \overline{Q^{n}} (see subsection 3.3), d\overline{Q^{n}} = Z_{T} dQ^{n}, h^{∗}(α) = (1−α)A^{∗}+ \frac{α(1−α)}{2}C^{∗}+\{α+(1−α)(1+x)−(1+x)^{1−α}\}∗ν^{∗}. It is easy to show using Itô’s formula that the Hellinger integrals H(α; Q^{∗}, \overline{Q^{∗}}), where d\overline{Q^{∗}} = Z_{T}^{∗}dQ^{∗}, satisfy

H(α; Q^{∗}, \overline{Q^{∗}}) = E(−h^{∗}(α))_{T} = exp (−h_{T}^{∗} (α)),  (6.6)

the last equality follows from stochastic continuity of X^{∗}. On the other hand, since ν({t} × ℝ) ≤ 1, it follows from (6.1) that

sup_{t≤T} |Δh_{t}^{n}(α)| ⟶^{Q^{n}} 0.  (6.7)

Combining (6.5)–(6.7), we get

E(−h^{n}(α))_{T} ⟶^{Q^{n}} E(−h^{∗}(α))_{T} = H(α; Q^{∗}, \overline{Q^{∗}})

for any α ∈ (0, 1). Following the same lines as in the proof of Theorem 3.1 in Jacod (1989), one can show that the last relation implies

lim_{n→∞} H(α; Q^{n}, \overline{Q^{n}}) = H(α; Q^{∗}, \overline{Q^{∗}}), α ∈ (0, 1),

and the claim follows from Proposition 2.5.

6.2. Let us fix some notation. We denote by 𝔻(ℝ^{d}) the Skorokhod space 𝔻([0, T], ℝ^{d}) of càdlàg functions [0, T] ∋ t ⇝ α(t) ∈ ℝ^{d}, equipped with the Skorokhod topology, D(ℝ^{d}) is the Borel σ-algebra in this space, {D_{t}(ℝ^{d})}_{0≤t≤T} is the filtration generated by the canonical process. The weak convergence in this space is denoted by ⟶^{L(𝔻^{d})}. If d = 1 then the index d is omitted in this notation. A sequence {P^{n}} of probability measures on (𝔻(ℝ^{d}), D(ℝ^{d})) is said to be C-tight if it is tight and any cluster point P of {P^{n}} is concentrated on the subspace of all continuous functions. Given a càdlàg stochastic process V = {V_{t}}_{0≤t≤T} with values in ℝ^{d}, we denote by L(V | P) its distribution in 𝔻(ℝ^{d}) relative to a measure P. Here we assume that Z^{∗} is a solution to the stochastic differential equation

dZ_{t}^{∗} = Z_{t}^{∗}σ(t, Z_{t}^{∗})dW_{t}, 0 ≤ t ≤ T, Z_{0}^{∗} = 1,

where W is a standard Wiener process. More precisely, we suppose that σ is a continuous bounded function, Ω^{∗} = 𝔻(ℝ), F_{T}^{∗} = D(ℝ), F_{t}^{∗} = D_{t}(ℝ), Z_{T}^{∗}(α) = α(t) is the canonical process, and Q^{∗} is the unique probability measure on (Ω^{∗}, F_{T}^{∗}) under which Q^{∗}(Z_{0}^{∗} = 1) = 1, Z is an a.s. continuous strictly positive martingale with the quadratic characteristic ∫_{0}^{t} (Z_{s−}^{∗})^{2}c(s, Z_{s−}) ds, where c(t, x) = σ^{2}(t, x). There are general theorems on the weak convergence of likelihood processes to the limit of our type, see Theorems X.1.59 and X.1.64 in Jacod and Shiryaev (1987) and Theorem 1 in Kramkov (1993), the latter deals even with a more general limiting model. But their assumptions are not well adapted to our case. We propose another result of this type.

Proposition 6.2 Under the above assumptions, let

sup_{t≤T} | \frac{1}{8}C_{T} + \frac{1}{2}(1 − \sqrt{1 + x})^{2} ∗ ν^{n} − \frac{1}{8} ∫_{0}^{t} c(s, Z_{s−}) ds | ⟶^{Q^{n}} 0,  (6.8)

and

|x|1_{{|x|>ε}} ∗ ν_{T}^{n} ⟶^{Q^{n}} 0  for all ε > 0.  (6.9)

Then L(Z_{T} | Q^{n}) ⇒ L(Z_{T}^{∗} | Q^{∗}).

Proof. Again, we give only an outline of the proof.

Since the function c is bounded, it follows from (6.8) that the sequences {L(C | Q^{n})} and {L((x^{2} ∧ |x|) ∗ ν^{n} | Q^{n})} are C-tight. Taking into account (3.7) and (6.9), we obtain from Theorem VI.4.18 in Jacod and Shiryaev (1987) that the sequence {L(X | Q^{n})} is tight in 𝔻(ℝ). Moreover, (6.9) implies that

sup_{t≤T} |ΔX_{t}| ⟶^{Q^{n}} 0,  (6.10)

hence the sequence {L(X | Q^{n})} is C-tight. Furthermore, Jacod’s condition

the sequence {L(Var(B^{n}(ϕ))_{T} | Q^{n})} is tight in ℝ  (6.11)

is satisfied due to (3.7) and (6.8). Let L(X | Q^{n_{k}}) ⟶^{L(𝔻)} L(X̂ | Q̂) for a subsequence {n_{k}}, where Q̂ is a probability measure on (𝔻(ℝ), D(ℝ)) and X̂ is the canonical process on 𝔻(ℝ). By Corollary 2.3 and Lemma 3.1 in Jakubowski et al. (1989), X̂ is a semimartingale on (𝔻(ℝ), D(ℝ), {D_{t}(ℝ))}_{0≤t≤T} , Q̂). Put Ẑ = E(X̂). Then Ẑ is a Q̂-a.s. strictly positive continuous process. In view of (6.11), we have

L(X, [X, X], Z | Q^{n_{k}}) ⟶^{L(𝔻^{3})} L(X̂, [X̂, X̂], Ẑ | Q̂),  (6.12)

see Theorem VI.6.1 in Jacod and Shiryaev (1987) and Proposition 6.1 in Mémin (1985), cf. also Jakubowski et al. (1989). Thus, the sequence {L(Z | Q^{n})} is tight in 𝔻(ℝ), and we only need to prove that L(Ẑ | Q̂) = Q^{∗} every time (6.12) holds. To simplify the notation we shall assume without loss of generality that the subsequence {n_{k}} coincides with the original sequence {n}. Denote Q′ = L(Ẑ | Q̂). Let us recall that the canonical process on 𝔻(ℝ) is denoted also by Z^{∗} so that

L(Z | Q^{n}) ⟶^{L(𝔻)} L(Z^{∗} | Q′) = Q′.

Put d\overline{Q^{n}} = Z_{T} dQ^{n} and let h^{n} be the Hellinger process of order 1/2 for Q^{n} and \overline{Q^{n}}. In view of (3.10), (6.8) can be rewritten in the form

sup_{t≤T} | h_{T}^{n} − \frac{1}{8} ∫_{0}^{t} c(s, Z_{s−}) ds | ⟶^{Q^{n}} 0.  (6.13)

Hence, the sequence {L(h^{n} | Q^{n})} is C-tight, in particular,

sup_{t≤T} |Δh_{t}^{n}| ⟶^{Q^{n}} 0.  (6.14)

It follows from (6.13) and (6.14) that

sup_{t≤T} | E(−h^{n})_{t} − exp(−h_{t}^{n})| ⟶^{Q^{n}} 0.  (6.15)

Put Y = \sqrt{Z} and y = \frac{1}{Y_{−}} · Y. Then

Δy = \sqrt{1 + ΔX} − 1,  (6.16)

in particular,

sup_{t≤T} |Δy_{t}| ⟶^{Q^{n}} 0.  (6.17)

Since E(X) = Z = Y^{2} = E(y)^{2}, by Yor’s formula

X = 2y + [y, y].  (6.18)

Put L^{n} = Y/E(−h^{n}). By Proposition V.4.16 in Jacod and Shiryaev (1987), L^{n} is a Q^{n}-local martingale. Since c is bounded, it follows from (6.13) and (6.14) that there is a δ > 0 such that lim_{n→∞} Q^{n}(E(−h^{n})_{T} ≤ δ) = 0. Then, similarly to the proof of Theorem 3.1 in Jacod (1989), one can construct a sequence {τ_{n}} of {F_{t}}-stopping times such that

lim_{n→∞} Q^{n}(τ_{n} < T) = 0 and E(−h^{n})_{τ_{n}} ≥ δ.  (6.19)

Since the family {L(Y_{τ_{n}∧t} | Q^{n})}_{0≤t≤T, n≥1} is uniformly integrable, the same is true for the family {L(L_{τ_{n}∧t}^{n} | Q^{n})}_{0≤t≤T, n≥1}; the same argument with an arbitrary stopping time τ instead of t shows that the stopped process {L_{τ_{n}∧t}^{n}}_{0≤t≤T} is a Q^{n}-martingale. Since

L_{τ_{n}∧t}^{n} − \sqrt{Z_{t}} exp (\frac{1}{8} ∫_{0}^{t} c(s, Z_{s−}) ds) ⟶^{Q^{n}} 0

for all t ∈ [0, T] due to (6.13), (6.15) and (6.19), we may apply Proposition IX.1.12 in Jacod and Shiryaev (1987) to obtain that L^{∗} := \sqrt{Z^{∗}} exp (\frac{1}{8} ∫_{0}^{t} c(s, Z_{s−}^{∗}) ds) is a martingale on (Ω^{∗}, F_{T}^{∗}, {F_{t}^{∗}}, Q′). In particular, Z^{∗} is a Q′-semimartingale. Put X^{∗} = \frac{1}{Z_{−}^{∗}} · Z^{∗}. The next step of the proof is to determine the quadratic characteristic [X^{∗}, X^{∗}] of X^{∗} under Q′. To proceed, put

m^{n} = y + h^{n}.  (6.20)

It follows from the general definition of the Hellinger process, see Chapter IV, § 1b in Jacod and Shiryaev (1987), that m^{n} is a Q^{n}-local martingale. Since X is a Q^{n}-local martingale, it is easy to show replacing y in (6.18) by its expression from (6.20), that m^{n} is a Q^{n}-locally square integrable martingale with the quadratic characteristic (relative to Q^{n})

⟨m^{n}, m^{n}⟩ = 2h^{n} − [h^{n}, h^{n}],  (6.21)

see also Mémin (1985). Let ν̃^{n} be the Q^{n}-compensator of the jump measure of m^{n}. It follows from (6.16) and (6.9) that the Lindeberg-type condition

x^{2}1_{{|x|>ε}} ∗ ν̃_{T}^{n} ⟶^{Q^{n}} 0 for all ε > 0  (6.22)

holds. Using (6.13), (6.21), (6.22) and some standard arguments, see e.g. the proof of Lemma 5 in § 5 of Chapter 5 in Liptser and Shiryaev (1989), one obtains

sup_{t≤T} |[m^{n}, m^{n}]_{t} − ⟨m^{n}, m^{n}⟩_{t}| ⟶^{Q^{n}} 0.

Combining this relation with (6.13), (6.14), (6.20) and (6.21), we get

sup_{t≤T} | [y, y]_{t} − \frac{1}{4} ∫_{0}^{t} c(s, Z_{s−}) ds | ⟶^{Q^{n}} 0.  (6.23)

Moreover, it can be easily seen from (6.18), (6.17) and (6.23) that

sup_{t≤T} |[X, X]_{t} − 4[y, y]_{t}| ⟶^{Q^{n}} 0,

hence

sup_{t≤T} | [X, X]_{t} − ∫_{0}^{t} c(s, Z_{s−}) ds | ⟶^{Q^{n}} 0.  (6.24)

Now we remark that, evidently,

L(X̂, [X̂, X̂], Ẑ | Q̂) = L(X^{∗}, [X^{∗}, X^{∗}], Z^{∗} | Q′).

Combining this statement with (6.12) and (6.24), we obtain

[X^{∗}, X^{∗}] = C^{∗} Q′-a.s., where C_{t}^{∗} = ∫_{0}^{t} c(s, Z_{s−}^{∗}) ds.

The rest is easy. The processes Z^{∗}, L^{∗} and C^{∗} are Q′-a.s. continuous, and it follows from the previous formula that

[Z^{∗}, Z^{∗}] = (Z^{∗})^{2} · C^{∗},  (6.25)

thus it remains to prove that Z^{∗} is a Q′-martingale. By Itô’s formula

Z^{∗} = (L^{∗})^{2}e^{−C^{∗}/4} = 2\frac{Z^{∗}}{L^{∗}} · L^{∗} − \frac{1}{4}Z^{∗} · C^{∗} + \frac{Z^{∗}}{(L^{∗})^{2}} · [L^{∗}, L^{∗}].  (6.26)

Taking the quadratic variations in (6.26), we obtain

[Z^{∗}, Z^{∗}] = 4 (\frac{Z^{∗}}{L^{∗}} )^{2} · [L^{∗}, L^{∗}].  (6.27)

Combining (6.25)–(6.27), we obtain

Z^{∗} = 2\frac{Z^{∗}}{L^{∗}} · L^{∗}.

Since L^{∗} is a Q′-martingale, Z^{∗} is a Q′-local martingale. The proper martingale property follows now from Novikov’s criterion, as σ is bounded.

7 Discrete time models

Let us consider a discrete time market model. For notational simplicity, we take B_{n} = 1 for n = 0, 1, …, N, but the arguments can be applied for any deterministic sequence {B_{n}}_{0≤n≤N}. We assume that the risky asset is modeled by

S_{0} = 1, S_{n} = ∏_{k=1}^{n}(1 + Y_{k}), n = 1, …, N,

for {Y_{k}}_{1≤k≤N} an adapted sequence of random variables on (Ω, F, F = {F_{t}}_{0≤t≤T} , P), with Y_{k+1} independent from F_{k} for each k, and such that Y_{k} > −1 for all k. Also for notational simplicity we assume P ∈ P. By independence, denoting μ_{k} the distribution of Y_{k} under P, we have

∫ (1 + x) μ_{k}(dx) = 1, k = 1, …, N.

In order to describe the upper and lower distributions, denote I_{k} = supp (μ_{k}) the support of μ_{k} and consider also for each k

−a_{k} = inf I_{k}, b_{k} = sup I_{k}, −α_{k} = sup I_{k} ∩ [−1, 0], β_{k} = inf I_{k} ∩ [0, ∞].

Observe that some b_{k} can take the value ∞. Consider now an auxiliary probability space (Ω^{0}, F^{0}, P^{0}) and define on it independent random variables Y_{1}^{∗}, …, Y_{N}^{∗} such that Y_{k}^{∗} = −a_{k} with probability one if a_{k} = b_{k} = 0 or b_{k} = ∞, and, otherwise,

Y_{k}^{∗} = \{ \begin{matrix}b_{k}, & with probability & \frac{a_{k}}{a_{k}+b_{k}} , \\ −a_{k}, & with probability & \frac{b_{k}}{a_{k}+b_{k}} . \end{matrix}

Similarly, let Y_{∗1}, …, Y_{∗N} be independent random variables on (Ω^{0}, F^{0}, P^{0}) such that Y_{∗k} = 0 with probability one if α_{k} = β_{k} = 0, and, otherwise,

Y_{∗k} = \{ \begin{matrix}β_{k}, & with probability & \frac{α_{k}}{α_{k}+β_{k}} , \\ −α_{k}, & with probability & \frac{β_{k}}{α_{k}+β_{k}} . \end{matrix}

Put S_{0}^{∗} = S_{∗0} = 1, S_{n}^{∗} = ∏_{k=1}^{n}(1 + Y_{k}^{∗}), S_{∗n} = ∏_{k=1}^{n}(1 + Y_{∗k}), n = 1, …, N, and denote by Q^{∗} (resp. Q_{∗}) the law of S_{N}^{∗} (resp. S_{∗N}) under P^{0}. It is clear that E_{P^{0}}S_{N}^{∗} ≤ E_{P^{0}}S_{∗N} = 1.

Proposition 7.1 Consider the discrete market model described above and a convex pay-off function f ∈ C such that f(0) + \frac{f(∞)}{∞} < ∞. Then

(a) The upper price is given by

ℂ^{∗} = sup_{Q∈P} E_{Q}f(S_{N}) = J_{f}(Q^{∗}) = E_{P^{0}}f(S_{N}^{∗}) + \frac{f(∞)}{∞}(1 − E_{P^{0}}S_{N}^{∗}).

(b) The lower price is given by

ℂ_{∗} = inf_{Q∈P} E_{Q}f(S_{N}) = J_{f}(Q_{∗}) = E_{P_{0}}f(S_{∗N}).

Remark 7.1 If the σ-algebras F_{n}^{0} are generated by the process S_{∗} = {S_{∗n}}_{0≤n≤N}, i.e. F_{n}^{0} = σ{S_{∗0}, …, S_{∗n}}, the corresponding market is no-arbitrage and complete and ℂ_{∗} is the fair price of the option f(S_{∗N}). Similarly, if F_{n}^{0} are generated by the price process S^{∗} = {S_{n}^{∗}}_{0≤n≤N} and all b_{k} are finite, then the corresponding market is no-arbitrage and complete and ℂ^{∗} is the fair price of the option f(S_{N}^{∗}).

Proof. First we verify that the bounds hold, i.e. for any Q ∈ P

J_{f}(Q_{∗}) ≤ E_{Q}f(S_{N}) ≤ J_{f}(Q^{∗}).  (7.1)

The arguments are similar to those used in the proof of Comparison Lemma. According to Corollary 2.1 it is sufficient to check (7.1) for a nonnegative non-increasing convex function f. Let μ be a probability measure on [−1, ∞) such that ∫ (1 + x) μ(dx) = 1 and supp (μ) ⊆ I_{n} = [−a_{n}, −α_{n}] ∪ [β_{n}, b_{n}]. Denote also by μ_{n}^{∗} and μ_{∗n} the distributions of Y_{n}^{∗} and Y_{∗n} respectively under P^{0}. Let us first show that

∫ g(1 + x) μ_{∗n}(dx) ≤ ∫ g(1 + x) μ(dx) ≤ ∫ g(1 + x) μ_{n}^{∗}(dx)  (7.2)

for any nonnegative non-increasing g ∈ C. The first inequality follows from Jensen’s inequality if α_{n} = β_{n} = 0, the second one is trivial if a_{n} = b_{n} = 0. In other cases denote by m(x) and M(x) the functions corresponding to the straight lines passing through the points (−α_{n}, g(1−α_{n})), (β_{n}, g(1+β_{n})) and (−a_{n}, g(1−a_{n})), (b_{n}, g(1+b_{n})) respectively (if b_{n} = ∞ put M(x) ≡ g(−a_{n})). Then

m(x) ≤ g(1 + x) ≤ M(x) μ-a.s.

and

m(x) = g(1 + x) μ_{∗n}-a.s., M(x) = g(1 + x) μ_{n}^{∗}-a.s.,

and (7.2) follows. Put

G^{∗}(n, z) = E_{P^{0}}f(z ∏_{k=n+1}^{N}(1 + Y_{k}^{∗})), G_{∗}(n, z) = E_{P^{0}}f(z ∏_{k=n+1}^{N}(1 + Y_{∗k})).

Let ν_{n}(ω, dy) be a regular conditional probability of Y_{n} given F_{n−1} under Q. Then, Q-a.s., ∫ (1+y) ν_{n}(ω, dy) = 1 (since E_{Q}(1+Y_{n} | F_{n−1}) = 1) and supp (ν_{n}(ω, dy)) ⊆ I_{n} (since Q ∼ P). Since G^{∗}(n, z) and G_{∗}(n, z) are convex and non-increasing in z, we obtain from (7.2) that Q-a.s.

E_{Q}(G^{∗}(n, S_{n}) | F_{n−1}) = ∫ G^{∗}(n, S_{n−1}(1 + z))ν_{n}(dz)
≤ ∫ G^{∗}(n, S_{n−1}(1 + z))μ_{n}^{∗}(dz) = G^{∗}(n − 1, S_{n−1}),

where the last equality follows from the independence of Y_{n}^{∗}:

G^{∗}(n − 1, z) = E_{P^{0}}E_{P^{0}}{f(z ∏_{k=n}^{N}(1 + Y_{k}^{∗})) | Y_{n}^{∗}} = E_{P^{0}}G(n, z(1 + Y_{n}^{∗})).

Hence G^{∗}(n, S_{n}) is a Q-supermartingale. Similarly, G_{∗}(n, S_{n}) is a Q-submartingale. It remains to note that G^{∗}(0, S_{0}) = E_{P^{0}}f(S_{N}^{∗}) = J_{f}(Q^{∗}), G_{∗}(0, S_{0}) = E_{P^{0}}f(S_{∗N}) = J_{f}(Q_{∗}), E_{Q}G^{∗}(N, S_{N}) = E_{Q}G_{∗}(N, S_{N}) = E_{Q}f(S_{N}) completing the first step. Now we check that the obtained bounds are attained. This will be done by finding sequences of measures in the set

M = {Q ∈ P: Y_{k} and F_{k−1} are independent with respect to Q, k = 1, …, n}.

(7.3)

Namely, assume that for all k and small ε > 0 we have constructed strictly positive functions h_{k,ε}(x) defined on the support of μ_{k} such that

∫ h_{k,ε}(x) μ_{k}(dx) = 1 and  ∫ (1 + x)h_{k,ε}(x) μ_{k}(dx) = 1.  (7.4)

Define Q_{ε} by

dQ_{ε} = ∏_{k=1}^{N} h_{k,ε}(Y_{k}) dP.

It is clear that Q_{ε} ∈ M. Now, if the distributions μ_{k,ε}, defined by μ_{k,ε}(dx) = h_{k,ε}(x) μ_{k}(dx), weakly converge to μ_{k}^{∗} (resp. μ_{∗k}) as ε → 0, then L(S_{N} | Q_{ε}) weakly converge to Q^{∗} (resp. Q_{∗}), and we have lim_{ε→0} E_{Q_{ε}}f(S_{N}) = J_{f}(Q^{∗}) (resp. = J_{f}(Q_{∗})) by Propositions 2.5 and 2.6. The required functions h_{k,ε} are constructed as follows (we omit the index k everywhere). In the upper bound case it is sufficient to consider the case a > 0 and b > 0. Put

h_{ε}(x) = A(ε)1_{[−a,−a+ε]}(x) + ε1_{(−a+ε,b−ε)}(x) + B(ε)1_{[b−ε,b]}(x),

where b − ε is replaced by ε^{−1} if b = ∞. The positive constants A(ε) and B(ε) are chosen in order to (7.4) be satisfied. This can be done if ε is small enough. It is clear that the family μ_{k,ε} is asymptotically tight as ε → 0 and the support of any cluster point is {−a} if b = ∞ or {−a, b} if b < ∞; moreover, in the second case (b < ∞) the mean of a cluster point is 0. Hence, μ_{k,ε} ⇒ μ_{k}^{∗} as ε → 0.

In the lower bound case the construction is similar if α_{k} > 0 and β_{k} > 0:

h_{ε}(x) = A(ε)1_{[−α−ε,α]}(x) + ε1_{[−a,−α−ε)∪(β+ε,b]}(x) + B(ε)1_{[β,β+ε]}(x),

where A(ε) and B(ε) are chosen in such a way that (7.4) holds. Finally, if α = β = 0 (but a > 0 and b > 0), we use the functions

h_{ε}(x) = A(ε)1_{[−a,−ε)}(x) + \frac{1 − ε}{μ([−ε, ε])}1_{[−ε,ε]}(x) + B(ε)1_{(ε,b]}(x).

If ε is small enough, then one can find positive A(ε) and B(ε) so that (7.4) is satisfied, concluding the proof.

A final remark, concerning more general discrete time market models (not necessarily satisfying our independence assumptions), is that if there exists a martingale measure equivalent to the initial measure P belonging to the set M in (7.3), then the results in Proposition 7.1 are still in force. Let us illustrate this remark with the conditionally Gaussian model considered in Shataev (1998). In this model, B_{n} = 1 for all n = 0, …, N, and a vector of independent random variables ε_{n} (n = 1, …, N) with standard Gaussian distribution is considered, F_{n} = σ{ε_{1}, …, ε_{n}}. The stock prices evolves according to S_{n} = S_{0}e^{h_{1}+...+h_{n}}, for each n = 1, …, N, where the conditionally Gaussian random variables h_{n} satisfy h_{n} = σ_{n}ε_{n}, with σ_{n} = σ_{n}(ε_{1}, …, ε_{n−1}), and σ_{n} are positive and continuous functions. The joint density of the vector (h_{1}, …, h_{N}) is given by

f(x_{1}, …, x_{N}) = ∏_{n=1}^{N} \frac{1}{\sqrt{2π}σ̃_{n}(x_{1}, …, x_{n−1})} exp (\frac{−x_{n}^{2}}{2σ̃_{n}^{2}(x_{1}, …, x_{n−1})}),

where σ̃_{n}(x_{1}, …, x_{n−1}) is defined by σ̃_{n}(h_{1}, …, h_{n−1}) = σ_{n}(ε_{1}, …, ε_{n−1}). A martingale measure Q in the set M can be then constructed with the density of the random vector (h_{1}, …, h_{N}) given by

f^{Q}(x_{1}, …, x_{N}) = ∏_{n=1}^{N} \frac{1}{\sqrt{2π}} exp ( − \frac{1}{2}(x_{n}^{2} − 1/2)^{2}).

Namely, put \frac{dQ}{dP} = \frac{f^{Q}(h_{1},...,h_{N})}{f(h_{1},...,h_{N})}. Therefore, conclusions of Proposition 7.1 hold. As the support of the distribution of the random variables Y_{n} = e^{h_{n}} − 1 is the set (−1, ∞), both the upper and lower universal bounds are attained.

8 Diffusion with jumps model

8.1. We now consider a continuous time model. For simplicity we take B = {B_{t}}_{0≤t≤T} with B_{t} = 1. In what respects the risky asset, denoted by Z = {Z_{t}}_{0≤t≤T} we assume that is the solution of the stochastic differential equation (s.d.e.)

dZ_{t} = Z_{t−}[a(t, Z_{t−})dt + σ(t, Z_{t−})dW_{t} + δ(t, Z_{t−}, x) ∗ (p(dt, dx) − q(dt, dx))], (8.1)

and Z_{0} = 1. The driving terms are

• W = {W_{t}}_{0≤t≤T} a standard Wiener process on B = (Ω, F_{T} , {F_{t}}_{0≤t≤T} , P), a stochastic basis;

• p a Poisson random measure on [0, T] × ℝ independent of W with intensity q(dt, dx) = dt K(dx), on the same stochastic basis B, K is a σ-finite measure on ℝ.

The coefficients a: [0, T] × (0, ∞) → ℝ, σ: [0, T] × (0, ∞) → (0, ∞), and δ: [0, T] × (0, ∞) × ℝ → (−1, ∞) are Borel functions, that satisfy:

(i) Local Lipschitz conditions: For each n ∈ ℕ there exist θ_{n} > 0 and ρ_{n}: ℝ → [0, ∞) satisfying ∫_{ℝ} ρ_{n}^{2}(x) K(dx) < ∞, such that for each t ∈ [0, T] and 0 < z, z′ ≤ n,

|za(t, z) − z′a(t, z′)| ≤ θ_{n}|z − z′|,  (8.2)
|zσ(t, z) − z′σ(t, z′)| ≤ θ_{n}|z − z′|,  (8.3)
|zδ(t, z, x) − z′δ(t, z′, x)| ≤ ρ_{n}(x)|z − z′|.  (8.4)

(ii) Boundness conditions: There exist positive constants a_{1}, σ_{0}, σ_{1} and ρ: ℝ → [0, ∞) satisfying ∫_{ℝ} ρ^{2}(x) K(dx) < ∞, such that for each t ∈ [0, T] and all z > 0,

|a(t, z)| ≤ a_{1},  (8.5)
0 < σ_{0} ≤ |σ(t, z)| ≤ σ_{1},  (8.6)
|δ(t, z, x)| ≤ ρ(x).  (8.7)

According to classical results on existence and uniqueness of solutions of s.d.e. (see for instance Jacod (1979), Theorem 14.23), conditions (8.2) to (8.7) ensure the existence of strong solutions of (8.1). Conditions (8.5), (8.6) and (8.7) are in fact slightly more stringent than usual conditions, and allow some useful developments.

We will need some additional assumptions: (iii) Conditions on volatility: The function σ(t, z) is continuous in (t, z). The function \frac{∂}{∂z}(zσ(t, z)) is continuous for all (t, z), and Lipschitz continuous and bounded in z ∈ (0, ∞) uniformly in t ∈ [0, T]. (iv) Conditions on jumps: In order to ensure the presence of jumps in the model, assume the existence of a positive constant c_{0} such that

∫_{[0,T]×ℝ} |δ(t, z, x)|dt K(dx) ≥ c_{0} > 0.  (8.8)

According to our previous framework, if we introduce X = {X_{t}}_{0≤t≤T} given by

X_{t} = ∫_{0}^{t} a(t, Z_{t−})dt + ∫_{0}^{t} σ(t, Z_{t−})dW_{t} + ∫_{[0,T]×ℝ} δ(t, Z_{t−}, x) ∗ (p(dt, dx) − q(dt, dx)),

(8.9)

then we can write Z = E(X). Observe that if p jumps at point (t, x), then ΔX_{t} = δ(t, Z_{t−}, x) > −1, so Z is strictly positive under P. Equation (8.9) is the canonical decomposition of X as a special semimartingale, and (as in Theorem III.2.26 in Jacod and Shiryaev (1987)) the triplet of predictable characteristics of X under P is

\{ \begin{matrix}A_{t} = ∫_{0}^{t} a(s, Z_{s−})ds, \\ C_{t} = ∫_{0}^{t} σ(s, Z_{s−})^{2}ds, \\ ν(ω, dt, B) = ∫_{ℝ} 1_{B\\{0\}}(δ(t, Z_{t−}, x))K(dx) dt. \end{matrix}  (8.10)

where B is a Borel set in ℝ. (Here and in (8.14) and (8.17) below it is more convenient to consider the predictable process A of bounded variation in the canonical decomposition of the special semimartingale X instead of the first characteristics B(ϕ); A and B(ϕ) are connected according to (3.13).) From this it follows that, for f: ℝ → ℝ, with f(0) = 0,

f(x) ∗ ν_{T} = ∫_{[0,T]×ℝ} f(δ(t, Z_{t−}, x))dt K(dx).  (8.11)

8.2. Now we construct a subset M of martingale measures (M ⊂ P) in the following way. Given constants H > −1 and ε ∈ (0, 1), consider a function Y = Y_{H,ε}: ℝ → (0, ∞) given by

Y (x) = 1 + H1_{{|x|>ε}}.  (8.12)

Note that by condition (8.7) and (8.11),

|x(Y (x) − 1)| ∗ ν_{T} = H|x|1_{{|x|>ε}} ∗ ν_{T} = H ∫_{[0,T]×ℝ} |δ(t, Z_{t−}, x)|1_{{|δ(t,Z_{t−},x)|>ε}}dt K(dx)

≤ HT ∫_{ℝ} ρ(x)1_{{ρ(x)>ε}}K(dx) ≤ ε^{−1}HT ∫_{ℝ} ρ^{2}(x)K(dx) < ∞.

Define β = β(t, z) by the equation

a(t, z) + β(t, z)σ^{2}(t, z) + ∫_{ℝ} δ(t, z, x)[Y (δ(t, z, x)) − 1]K(dx) = 0.  (8.13)

Then

β(t, z) = \frac{−1}{σ^{2}(t, z)}[a(t, z) + H ∫_{ℝ} δ(t, z, x)1_{\{|δ(t,z,x)|>ε\}}K(dx)].

Therefore, given H and ε, the function β is uniformly bounded,

|β(t, z)| ≤ \frac{1}{σ_{0}^{2}} [a_{1} + H ∫_{ℝ} ρ(x)1_{\{ρ(x)>ε\}}K(dx)] =: β_{1}.

Now we can define the local martingale N = {N_{t}}_{0≤t≤T} by

N_{t} = ∫_{0}^{t} β(s, Z_{s−})σ(s, Z_{s−}) dW_{s} + ∫_{[0,t]×ℝ}(Y (x) − 1) d(μ^{X} − ν),

with μ^{X} the jump measure of the process X, and consider D = E(N). As ΔN = H1_{{|ΔX|>ε}}, for f: ℝ → ℝ such that f(0) = 0, and ν^{N} the compensator under P of the jump measure of the process N, we have

f(x) ∗ ν_{T}^{N} = f(H) ∫_{[0,T]×ℝ} 1_{{|δ(t,Z_{t−},x)|>ε}}dt K(dx).

Then we can estimate

⟨N^{c}, N^{c}⟩_{T} + \frac{x^{2}}{1 + |x|} ∗ ν_{T}^{N} ≤ T β_{1}^{2}σ_{1}^{2} + T \frac{H^{2}}{1 + H} K(\{x: ρ(x) > ε\}) < ∞,

and condition (3.8) holds, giving that D is in fact a martingale. This allows us to introduce the probability measure Q on (Ω, F_{T}) by the relation

\frac{dQ}{dP} = D_{T} .

Now we calculate the triplet of predictable characteristics of X under Q. It follows from the construction of D and from Girsanov’s Theorem, see Proposition III.3.24 in Jacod and Shiryaev (1987), that the Girsanov coefficients β^{Q} and Y^{Q} for X, see (3.5), satisfy β_{t}^{Q} = β(t, Z_{t−}) and Y^{Q} = Y. Using (3.5), (3.13), (8.10) and (8.13), we obtain that the predictable characteristics of X under Q are

\{ \begin{matrix}A_{t}^{Q} = 0, \\ C^{Q} = C, \\ ν^{Q} = Y ν. \end{matrix}  (8.14)

Therefore, X is a local martingale under Q. By (8.11) and (8.14)

⟨X^{c}, X^{c}⟩_{T} + \frac{x^{2}}{1 + |x|} ∗ ν_{T}^{Q} ≤ T σ_{1}^{2} + (1 + H)T ∫_{ℝ} ρ^{2}(x) K(dx) < ∞,

and condition (3.8) holds, so Z = E(X) is a uniformly integrable Q-martingale. This shows that the set P is not empty.

8.3. Consider a convex function f that satisfies the finiteness condition (1.4). In order to determine a lower bound for option prices in the model just introduced, consider the process Z^{∗} = {Z_{t}^{∗}}_{0≤t≤T} defined as the solution of the s.d.e.

dZ_{t}^{∗} = Z_{t}^{∗}σ(t, Z_{t}^{∗})dW^{∗},  0 ≤ t ≤ T,  Z_{0}^{∗} = 1,  (8.15)

with σ introduced in (8.1), and W^{∗} a standard Wiener process on a stochastic basis B^{∗} = (Ω^{∗}, F_{T}^{∗}, {F_{t}^{∗}}_{0≤t≤T} , Q^{∗}). Define X^{∗} = {X_{t}^{∗}}_{0≤t≤T} by

X_{t}^{∗} = ∫_{0}^{t} σ(s, Z_{s}^{∗})dW_{s}^{∗}.  (8.16)

For each f ∈ C_{0}^{2} consider

G(t, z) = E_{Q^{∗}}(f(Z_{T}^{∗})|Z_{t}^{∗} = z).

Observe that, conditions (B) and (C) in the comparison Lemma hold, as discussed in 5.2.2.

8.4. Let now Q be an arbitrary martingale measure i.e. Q ∈ P. The process Z is a positive Q-martingale, and the representation Z = E(X) holds under Q. Using (3.5) and the fact that X is a local martingale under Q, we obtain that the triplet of predictable characteristics of X under Q has the form

\{ \begin{matrix}A^{Q} = 0, \\ C_{t}^{Q} = C_{t} = ∫_{0}^{t} σ^{2}(s, Z_{s−})ds, \\ ν^{Q}(ω, dt, B) = ∫_{B} Y^{Q}(ω, t, x)ν(ω, dt, dx) \\ = ∫_{ℝ} 1_{B\\{0\}}(δ(t, Z_{t−}, x))Y^{Q}(ω, t, δ(t, Z_{t−}, x))K(dx)dt, \end{matrix}  (8.17)

where Y^{Q} is a nonnegative predictable function. This shows that decomposition (5.3) and condition (A) in the comparison Lemma hold. Observe finally, that the triplet for X^{∗} in (8.16) under Q^{∗} is (0, C^{∗}, 0), where C^{∗} satisfies (5.4) and (5.5) with c^{∗} = σ^{2}. So conditions (5.10), (5.11) and (5.12) hold (with the opposite inequalities). We are then in position to apply the Comparison Lemma 5.1 for the process Z on B = (Ω, F, {F_{t}}_{0≤t≤T} , Q) and Z^{∗} on B^{∗} = (Ω^{∗}, F_{T}^{∗}, {F_{t}^{∗}}_{0≤t≤T} , Q^{∗}), obtaining

E_{Q}f(Z_{T}) ≥ E_{Q^{∗}}f(Z_{T}^{∗}).

In order to conclude that ℂ_{∗} = E_{Q^{∗}}f(Z_{T}^{∗}), we consider the question of attainability of lower bounds. The idea is to construct a sequence of martingale measures Q^{n} in M ⊂ P as in subsection 8.2 that verify conditions (6.8) and (6.9) in Proposition 6.2. Take two sequences {ε_{n}}, {H_{n}}, with 0 < ε_{n} < 1, H_{n} > −1, and ε_{n} → 0, 1+H_{n} → 0. Define, for each n, Y^{n} = Y_{H^{n},ε^{n}} as in (8.12). Define, also for each n, β^{n} by formula (8.13), with Y^{n} instead of Y. As done in subsection 8.2, we construct Q^{n}. Triplets of X under Q^{n} are denoted by (0, C, ν^{n}), as in (8.14). We verify (6.8). As C_{t} = ∫_{0}^{t} σ^{2}(s, Z_{s−})ds, this condition reduces to the verification of

(1 − \sqrt{1 + x})^{2} ∗ ν_{T}^{n} ⟶^{Q^{n}} 0.

Now,

(1 − \sqrt{1 + x})^{2} ∗ ν_{T}^{n} ≤ (1 − \sqrt{1 + x})^{2}1_{\{|x|≤ε_{n}\}} ∗ ν_{T}  (8.18)
+ (1 + H_{n})(1 − \sqrt{1 + x})^{2}1_{\{|x|>ε_{n}\}} ∗ ν_{T} .

The second term in (8.18) vanishes by the dominated convergence theorem, as on the one hand, the integrand goes to zero, and on the other hand,

(1 − \sqrt{1 + x})^{2}1_{\{|x|≤1\}} ∗ ν_{T} ≤ x^{2} ∗ ν_{T} = ∫_{[0,T]×ℝ} δ^{2}(t, Z_{t−}, x)dt K(dx)
≤ T ∫_{ℝ} ρ^{2}(x) K(dx).

Similar computations give

(1 + H_{n})(1 − \sqrt{1 + x})^{2}1_{\{|x|>ε_{n}\}} ∗ ν_{T} ≤ (1 + H_{n})x^{2} ∗ ν_{T}
= (1 + H_{n}) ∫_{[0,T]×ℝ} δ^{2}(t, Z_{t−}, x)dt K(dx) ≤ (1 + H_{n})T ∫_{ℝ} ρ^{2}(x) K(dx) → 0,

if n → ∞, and (6.8) holds. Observe that in this particular case, (6.9) follows from (6.8). In this way, we conclude that the lower bound given by the Comparison Lemma is attained, and in consequence

ℂ_{∗} = E_{Q^{∗}}f(Z_{T}^{∗}).

8.5. We finally verify that the upper bound is trivial, i.e. the upper universal bound is attained. Consider the same sequence of functions Y^{n} as above, but take H_{n} arbitrary by the moment, and ε_{n} → 0 as formerly. We verify that the corresponding Hellinger processes satisfy condition (iii) in Proposition 4.1. Denote φ(x) = \frac{1}{2}(1 − \sqrt{1 + x})^{2}, and note that for each ε_{n} there exists a_{n} > 0 such that

φ(x)1_{{|x|>ε_{n}}} ≥ a_{n}|x|1_{{|x|>ε_{n}}}.

The Hellinger process of order \frac{1}{2} in (3.10), in this case satisfy

h^{n}(\frac{1}{2})_{T} = \frac{1}{8}C_{T} + \frac{1}{2}(1 − \sqrt{1 + x})^{2} ∗ ν_{T}^{n} ≥ a_{n}|x|1_{\{|x|>ε_{n}\}} ∗ ν_{T}^{n}

= a_{n}(1+H_{n})|x|1_{{|x|>ε_{n}}}∗ν_{T} = a_{n}(1+H_{n}) ∫_{[0,T]×ℝ} |δ(t, Z_{t−}, x)|1_{{|δ(t,Z_{t−},x)|>ε_{n}}}dt K(dx).

If we choose {H_{n}} such that a_{n}(1 + H_{n}) → ∞, as

lim_{n} ∫_{[0,T]×ℝ} |δ(t, Z_{t−}, x)|1_{{|δ(t,Z_{t−},x)|>ε_{n}}}dt K(dx) = ∫_{[0,T]×ℝ} |δ(t, Z_{t−}, x)|dt K(dx) ≥ c_{0}

by condition (8.8), we obtain that for any N > 0

lim_{n→∞} Q^{n}(h^{n}(\frac{1}{2})_{T} > N) = 1

and the proof of the upper bound is complete. In conclusion we formulate our result.

Proposition 8.1 Consider a model of a financial market with B = {B_{t}}_{0≤t≤T} such that B_{t} = 1, and Z = {Z_{t}}_{0≤t≤T} the solution of the s.d.e. given in (8.1). Assume that conditions (i), (ii), (iii) and (iv) in subsection 8.1 are fulfilled. Then for a convex payoff function f satisfying the finiteness condition (1.4) we have the following bounds on option prices.

(a) The upper price is given by the universal bound, i.e.

ℂ^{∗} = sup_{Q∈P} E_{Q}f(Z_{T}) = f(0) + \frac{f(∞)}{∞}.

(b) The lower price is given by a pure diffusion model with the same volatility structure, i.e.

ℂ_{∗} = inf_{Q∈P} E_{Q}f(Z_{T}) = E_{Q^{∗}}f(Z_{T}^{∗}),

where Z^{∗} is the solution of the s.d.e. (8.15).

9 Conclusions

Consider a general semimartingale model of a security market with finite horizon T, risk-less asset B and risky asset S. Given a convex function g = g(x) consider a European option with payoff g(S_{T}). In the presented paper, we study the problem of the determination of all possible prices for this option, obtained as expectations of the discounted payoff with respect to a generic probability measure that ranges along the set of all equivalent martingale measures. In other words, our purpose is the determination of the set of admissible prices for this option. As this set is an interval, our task is fulfilled in two steps: (i) the comparison of option prices under different martingale measures, more precisely, we give criteria to know whether the prices of European options with the same payoff function g and different stocks can be compared, and (ii) the convergence of option prices corresponding to a sequence of martingale measures, in order to verify optimality of the bounds obtained using (i). The main idea in (i) is to introduce a partial order in the set of (one-dimensional) probability distributions of discounted stock prices at the exercise time T, i.e. the law of the random variable Z_{T} := S_{T} /B_{T} under an arbitrary equivalent martingale measure. This partial order (adapted from the theory of statistical experiments) allows to find certain extremal measures, not necessarily in the original set (in many cases they are singular martingale measures). These measures give upper and lower bounds for the range of options prices. Weak convergence of probability measures is used in (ii), in order to determine whether the mentioned bounds are optimal, i.e. whether the obtained bounds are the supremum and infimum of the set of admissible option prices. Furthermore, the presence of the filtration makes possible to appeal to the theory of stochastic calculus for semimartingales. In reference to (i) we obtain predictable criteria to compare option prices (more precisely we represent Z^{i} = E(X^{i}) for i = 1, 2, and give a comparison result that involves the comparison (in certain sense) of the predictable local characteristics of X^{1} and X^{2} under the corresponding martingale measures). Particular attention is given to the necessity of the hypothesis in this comparison result. This question is answered, constructing a counterexample of two Lévy driven stocks (one driven by a standard Wiener processes, and the other by the compensated difference of two Poisson processes), with comparable prices for all convex functions for some fixed exercise time, but predictable characteristics not comparable in the mentioned sense. In what respects (ii) two types of results are discussed. First, the case were the universal (i.e. that hold a priori for any martingale measure) bounds are attained, obtaining predictable criteria in terms of the Hellinger process corresponding to the discounted stock price. In second place results about weak convergence of the discounted stock prices in terms of the predictable characteristics of their stochastic logarithms are given in two situations: when the process X in the limit is a process with independent increments, and when the limit is a stochastic volatility model. Special attention is given to random walk models, and models where the stock satisfies a stochastic differential equation with jumps. In both cases complete solution to the problem of determination of the range of option prices is given, under mild assumptions. Finally, it is interesting to remark that the partial ordering introduced in (i) is strongly related to the comparison of binary statistical experiments, where the discounted stock process Z plays the role of the density process of the statistical experiment. Part (ii) can be seen as weak convergence of the corresponding density process. This analogy, from one side, helped to understand the above mentioned problems, and, on the other, gives the possibility of interpreting our results in terms of the theory of statistical experiments.

This research was begun in autumn 1999 during the visit of the first author to Montevideo. He would like to express his gratitude to PEDECIBA-Matemática, Centro de Matemática, Facultad de Ciencias for their hospitality.

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