Perpetual Options for Lévy Processes in the Bachelier Model

Ernesto Mordecki ^{∗} Facultad de Ciencias. Centro de Matemática. Iguá 4225. CP 11400. Montevideo. Uruguay.

October 30, 2000

Abstract

Solution to the optimal stopping problem

V (x) = sup_{τ} Ee^{−δτ} g(x + X_{τ})

is given, where X = {X_{t}}_{t≥0} is a Lévy process, τ is an arbitrary stopping time, δ ≥ 0 is a discount rate, and the reward function g takes the form g_{c}(x) = (x−K)^{+} or g_{p}(x) = (K−x)^{+} Results, interpreted as option prices of perpetual options in Bachelier’s model are expressed in terms of the distribution of the overall supremum in case g = g_{c} and overall infimum in case g = g_{p} of the process X killed at rate δ. Closed form solutions are obtained under mixed exponentially distributed positive jumps with arbitrary negative jumps for g_{c}, and under arbitrary positive jumps and mixed exponentially distributed negative jumps for g_{p}. In case g = g_{c} a prophet inequality comparing prices of perpetual look-back call options and perpetual call options is obtained.

Keywords and Phrases: Optimal stopping, Lévy process, mixtures of exponential distributions, perpetual American options, prophet inequalities

Mathematics Subject Classification (1991): 60G40, 60J30, 62L15, 90A09.

1 Introduction and general results

1.1 Lévy processes

Let X = {X_{t}}_{t≥0} be a real valued stochastic process defined on a stochastic basis (Ω, F, F = (F_{t})_{t≥0}, P) that satisfy the usual conditions. Assume that X is càdlàg, adapted, X_{0} = 0, and for 0 ≤ s < t the random variable X_{t} − X_{s} is independent of the σ-field F_{s} with a distribution that only depends on the difference t − s. X is a process with stationary independent increments (PIIS), or a Lévy process. We will also consider the process X̂ = {X̂_{t}}_{t≥0} = {−X_{t}}_{t≥0}, called the dual process.

∗mordecki@cmat.edu.uy, http://www.cmat.edu.uy/˜ mordecki

If q ∈ R, Lévy-Khinchine formula states

Ee^{iqX_{t}} = exp \{t[ibq − \frac{1}{2}σ^{2}q^{2} + ∫_{R}(e^{iqy} − 1 − iqy1_{\{|y|<1\}})Π(dy)]\}, (1)

where b and σ ≥ 0 are real constants, and Π is a positive measure on R − {0} such that ∫ (1 ∧ y^{2})Π(dy) < +∞, called the Lévy measure. These parameters completely determine the law of the process. We always assume that the process does not degenerate, i.e. σ ≠ 0 or Π ≠ 0.

In order to use indistinctly Fourier or Laplace transforms, consider the set

C_{0} = {c ∈ R: ∫_{{|y|>1}} e^{cy}Π(dy) < ∞}. (2)

C_{0} is a convex set that contains the origin, and consists of all c ∈ R such that Ee^{cX_{t}} < ∞ for some t > 0. Furthermore, if z ∈ C and ℜ(z) ∈ C_{0} we can define the characteristic exponent

ψ(z) = bz + \frac{1}{2}σ^{2}z^{2} + ∫_{R}(e^{zy} − 1 − izy1_{\{|y|<1\}})Π(dy) (3)

having E|e^{zX_{t}}| < ∞ for all t ≥ 0, and

Ee^{zX_{t}} = e^{tψ(z)}.

Observe that if z = iq the preceding formula gives (1). See Sato (1999) for details.

When considering X as a semimartingale, denote by (bt, σ^{2}t, Π(dy)dt), or for short (b, σ^{2}, Π) the triplet of predictable characteristics of X (see II.4.19 in Jacod and Shiryaev (1987)). If we denote by (b̂, σ̂^{2}, Π̂) the triplet of X̂, the dual process, we have b̂ = −b, σ̂ = σ, and Π̂(dx) = Π(−dx). Given X and δ ≥ 0 denote

(4) M = sup_{0≤t<τ(δ)} X_{t} and I = inf_{0≤t<τ(δ)} X_{t},

where τ(δ) is an exponential random variable with parameter δ > 0, independent of X, and τ(0) = ∞. M and I will be called the supremum and infimum of X in both cases δ = 0 and δ > 0. τ is a stopping time relative to F if

τ: Ω → [0, +∞] and {τ ≤ t} ∈ F_{t} for all t ≥ 0.

Denote by M the class of all stopping times relative to F.

For general reference on the subject see Jacod and Shiryaev (1987), Skorokhod (1991), Bertoin (1996), or Sato (1999).

1.2 General results on optimal stopping

Consider the following problem: given a Borel function g: R → R, the reward function, a process X as above, and a discount rate δ ≥ 0, find a real function V and a stopping time τ^{∗} such that

V (x) = sup_{τ∈M} Ee^{−δτ} g(x + X_{τ}) = Ee^{−δτ^{∗}}g(x + X_{τ^{∗}}). (5)

V is called the cost function, and τ^{∗}, the stopping time that realizes the supremum, the optimal stopping time. We assume that e^{−δτ} g(x + X_{τ})1_{{τ=∞}} = lim sup_{t→∞} e^{−δt}g(x + X_{t}).

In the present paper solution to the problem (5) is given, when considering reward functions

g_{c}(x) = (x − K)^{+} and g_{p}(x) = (K − x)^{+} (6)

(with x^{+} = max(x, 0)) in terms of the distribution of the random variables M and I in (4). c and p stand for call and put options, and we interpret our results as the pricing of perpetual call and put options in the Bachelier (1900) model. In Section 2 closed form solutions are obtained for the call (respectively put) options, assuming that positive (resp. negative) jumps of X are distributed as a mixture of exponentials, and negative jumps (resp. positive) behave arbitrarily. In Section 3 we present a prophet inequality comparing the cost of a perpetual call and the expectation of the overall maximum of the process, i.e. the price of a perpetual look-back option. Section 4 contains the proofs, and Section 5 a conclusion. The presented results where partially announced in (Mordecki, (2000a)) were other examples are considered, and are complemented by the ones contained in the paper “Optimal Stopping and Perpetual Options for Lévy processes”, (Mordecki, (2000b)), where similar results for reward functions (e^{x}− K)^{+} and (K − e^{x})^{+} are obtained, and applications to finance are discussed in detail.

Theorem 1 Let X be a Lévy process and δ ≥ 0. Denote M and I as in (4)

(a) If EM < ∞, the solution to the problem (5) with reward g_{c}(x) = (x−K)^{+} has cost function V_{c}(x) = E(x + M − EM − K)^{+} (7)

and optimal stopping time

τ_{c}^{∗} = inf{t ≥ 0: x + X_{t} ≥ K + EM}.

(b) If EI > −∞, the solution to the problem (5) with reward g_{p}(x) = (K − x)^{+} has cost function

V_{p}(x) = E(K + EI − x − I)^{+}

and optimal stopping time

τ_{p}^{∗} = inf{t ≥ 0: x + X_{t} ≤ K + EI}.

Remark. As can bee seen in the proof, (b) follows easily from (a), considering the dual process. For this reason consider mainly the call case (a).

Theorem 1 essentially says, that although in general, the optimal stopping problem for a Markov process leds to a free boundary problem (Theorem III§8.15 in Shiryaev (1978)), in the case considered, this problem reduces to finding the distribution of M, a problem that has the advantage of having fixed boundary conditions. The conclusion is, that for a Lévy process, the optimal stopping problem with reward x^{+} can be explicitly solved once the distribution of M is known. The case of a random walk with δ = 0 was considered by Darling et al. (1972).

2 Closed solutions for mixed-exponential jumps

We specify now the Lévy measure in (1) in order to obtain closed solutions for the optimal stopping problem. Given a = (a_{1}, …, a_{n}) with ∑_{k=1}^{n} a_{k} = 1, a_{k} > 0 for k = 1, …, n, and α = (α_{1}, …, α_{n}), with 0 < α_{1} < α_{2} < … < α_{n}, denote by

h_{n}(y; a, α) = ∑_{k=1}^{n} a_{k}α_{k}e^{−α_{k}y}, y ≥ 0,

the density of a mixture of n exponential random variables with parameters α_{1}, …, α_{n} and mixture coefficients a_{1}, …, a_{n}.

Consider a Lévy process X with triplet (b, σ^{2}, Π), and Lévy measure given by

Π(y) = \{ \begin{matrix}λh_{n}(y; a, α)dy & y > 0, \\ π(dy) & y < 0, \end{matrix} (8)

where π(dy) is an arbitrary Lévy measure with support on (−∞, 0), and a and α are as before. X has mixed-exponentially distributed positive jumps. Its dual process X̂ has mixed exponentially distributed negative jumps and arbitrary positive jumps.

In this case, the set C_{0} in (2) contains the interval [0, α_{1}) and the characteristic exponent

ψ(z) = bz + \frac{1}{2}σ^{2}z^{2} + ∫_{−∞}^{0}(e^{zy} − 1 − zy1_{\{−1<y<0\}})π(dy)

+λ ∫_{0}^{∞}(e^{zy} − 1 − zy1_{{0<y<1}})h_{n}(y; a, α)dy

has an analytical continuation to the set {z ∈ C: ℜ(z) ≥ 0, z ≠ α_{k}, k = 1, …, n} given by

Ψ(z) = az + \frac{1}{2}σ^{2}z^{2} + ∫_{−∞}^{0}(e^{zy} − 1 − zy1_{\{−1<y<0\}})π(dy)+

+λ ∑_{k=1}^{n} a_{k} \frac{z}{α_{k} − z} , (9)

with

a = b − λ ∑_{k=1}^{n} \frac{a_{k}}{α_{k}} [1 − (1 + α_{k})e^{−α_{k}}].

The distribution of the random variable M in (4) for (b, σ^{2}, Π) as above with σ > 0, under either the condition δ > 0, or

ψ′(0+) = lim_{p→0+} \frac{ψ(p)}{p} = a + ∫_{−∞}^{−1} yπ(dy) + λ ∑_{k=1}^{n} \frac{a_{k}}{α_{k}} < 0, (10)

if δ = 0, where the integral can take the value −∞, is also a mixture of exponentials, and has density

f_{M}(y) = h_{n+1}(y; A, p) (11)

where p = (p_{1}, …, p_{n+1}), with

0 < p_{1} < α_{1} < p_{2} < … < α_{n} < p_{n+1} (12)

are the real and positive roots of Ψ(p) = δ and A = (A_{1}, …, A_{n+1}) are given by

A_{j} = \frac{∏_{k=1}^{n}(\frac{p_{j}}{α_{k}} − 1)}{∏_{k=1,k≠j}^{n+1}(\frac{p_{j}}{p_{k}} − 1)} j = 1, …, n + 1, (13)

as follows form Theorem 1.1 in Mordecki (1999). This gives the following result.

Theorem 2 Let X be a Lévy process with σ > 0, Lévy measure given by (8), and δ ≥ 0. Under the condition δ > 0 or (10) when δ = 0, we have

EM = ∑_{k=1}^{n+1} \frac{A_{k}}{p_{k}} ,

with p_{k}, k = 1, …, n + 1, the positive roots of Ψ(p) = δ, that satisfy (12), and A_{k} given in (13). Furthermore, the optimal stopping problem for X with reward g_{c}(x) = (x − K)^{+} has cost function

V_{c}(x) = \{ \begin{matrix}∑_{k=1}^{n+1} \frac{A_{k}}{p_{k}} e^{−p_{k}(x^{∗}−x)} & x ≤ x^{∗}, \\ x − K & x > x^{∗}, \end{matrix}

with x^{∗} = K + EM, and optimal stopping time

τ^{∗} = inf{t ≥ 0: x + X_{t} ≥ x^{∗}}.

Remarks. (1) In order to obtain closed solution for put options, we must specify the negative jumps of the Lévy process. Given X with Lévy measure in (8) the dual process X̂ has negative mixed exponentially distributed negative jumps, and it is direct (see proof of (b) for Theorem 1), under the same assumptions, denoting x_{p}^{∗} = K + EÎ

sup_{τ} E(K − x − X̂_{τ})^{+} = \{ \begin{matrix}K − x & x ≤ x_{p}^{∗}, \\ ∑_{k=1}^{n+1} \frac{A_{k}}{p_{k}} e^{−p_{k}(x_{p}^{∗}−x)} & x ≤ x_{p}^{∗}, \end{matrix}

and the optimal stopping time is

τ_{p}^{∗} = inf{t ≥ 0: x + X_{t} ≤ x_{p}^{∗}}.

(2) The results presented extend those in Taylor (1968) for the Wiener process, see also Shiryaev (1978), Mordecki (1998) for a compound Poisson process with exponential jumps, and Mordecki (1997) for a diffusion with exponential jumps. All these cases considered only δ = 0.

3 A prophet inequality for Lévy processes

Our final result states a prophet inequality for the class of Lévy processes. Recall that if K = 0 and g = g_{g} the cost function is given by

V (x) = sup_{τ} Ee^{−δτ} (x + X_{τ})^{+}.

Theorem 3 Let X be a Lévy process with EM < ∞, and x ≥ 0. Then, the following inequality holds:

V (x) ≤ E(x + M) ≤ eV (x).

This result generalizes the one obtained in Darling et al. (1972) for random walks, x = 0 and δ = 0. It is called a prophet inequality in the sense that compares the expected reward of a gambler that has complete foresight, or a prophet that can choose the moment at which the maximum is attained, obtaining x + M, with the expectation of the best possible strategy in the class M. For references on analogous results for sequences of random variables see Hill and Kertz (1990). For related results on prophet inequalities in finance see Kertz (1999).

4 Proofs

The proof of Theorem 1 is based on a result on random walks of independent interest, a slightly generalization of Darling et al. (1972).

Introduce a sequence X, X_{1}, X_{2}, … of i.i.d. random variables. As usual, partial sums are denoted by

S_{0} = 0, S_{n} = ∑_{k=1}^{n} X_{k}, n = 1, 2, …,

and τ denotes a stopping time with respect to the filtration generated by the random walk S = {S_{n}}_{n∈N}. The optimal stopping problem for this random walk with reward g and discount rate α ∈ (0, 1] consists in finding a real function V and a stopping time τ^{∗} such that

V (x) = sup_{τ} Eα^{τ} g(x + S_{τ}) = Eα^{τ^{∗}}g(x + S_{τ^{∗}}). (14)

Let τ(α) be a geometric random variable with P(τ(α) > k) = α^{k}, for k = 1, 2, … , independent of S with α ∈ (0, 1), and denote τ(1) = ∞. The supremum of the random walk killed at rate α is

W = inf_{0≤n<τ(α)} S_{n}. (15)

Proposition 1 Consider a random walk S, τ(α) and W as above. Assume EW < ∞ and denote

C(x) = E(x + W − EW)^{+}, σ^{∗} = inf{n ≥ 0: x + S_{n} ≥ EW }.

Then we have

(i) C(x) ≥ x^{+},

(ii) C(x) ≥ αEC(x + X),

(iii) C(x) = Eα^{σ^{∗}}(x + S_{σ^{∗}})^{+}.

and in consequence,

sup_{τ} Eα^{τ} (x + S_{τ})^{+} = E(x + W − EW)^{+} = Eα^{σ^{∗}}(x + S_{σ^{∗}})^{+}. (16)

Remark. For α = 1 this result is included in Darling et al. (1972).

Proof. (i) follows by Jensen’s inequality. To see (ii) introduce the random variable J, independent of S and τ(α), such that P(J = 0) = 1 − α, P(J = 1) = α. We claim

W =^{d} J(X + W)^{+}.

In fact,

Ee^{zJ(X+W)^{+}} = 1 − α + αEe^{z(X+W)^{+}}

= Ee^{zW} 1_{{τ(α)=1}} + ∑_{k=1}^{∞} αE(e^{z sup_{0≤n<k} S_{n}}1_{{τ(α)=k}}) = Ee^{zW} .

Now

C(x) = E(x + J(X + W)^{+} − EW)^{+} ≥ αE(x + X + W − EW)^{+} = αEC(x + X).

In order to see (iii), observe that {σ^{∗} < τ(α)} = {x + W ≥ EW }, and

E(x + W)1_{{x+W ≥EW }} = E(x + S_{σ^{∗}})1_{{x+W ≥EW }} + E(W − S_{σ^{∗}})1_{{x+W ≥EW }} =

Eα^{σ^{∗}}(x + S_{σ^{∗}})^{+} + EMP(σ^{∗} < τ(α))

and from this

Eα^{σ^{∗}}(x + S_{σ^{∗}})^{+} = E(x + W − EW)^{+}.

This proves (i), (ii) and (iii). The final conclusion (16) follows from Darling et al. (1972).

Proof of Theorem 1. First observe that (b) follows from (a).

sup_{τ} E(K − x − X_{τ})^{+} = sup_{τ} E(K − x + X̂_{τ})^{+}

= E(K − x + M̂ − EM̂)^{+} = E(K + EI − x − I)^{+},

where X̂ is the dual process, and M̂ = sup_{0≤t<τ(δ)} X̂_{t} = −I.

Let us now see (a). Without loss of generality we take K = 0, so g_{c}(x) = x^{+}. First we verify that (7) holds. In order to see this, on one side we recall the fact that the cost function of the optimal stopping problem considered is the minimal δ-excessive majorant function of the reward g_{c}, and satisfies

V_{c}(x) = lim_{n→∞} lim_{N→∞} Q_{n}^{N}g_{c}(x) (17)

where the operator Q_{n} is given by

Q_{n}g_{c}(x) = max{g_{c}(x), e^{−δ2^{−n}}Eg_{c}(x + X_{2^{−n}})},

and Q_{n}^{N} is the N^{th} iteration of Q_{n}. (see Shiryaev (1978) and VIII.2.4 in Shiryaev (1999)). Fix now n. The limit lim_{N→∞} Q_{n}^{N}g_{c}(x) is the cost function for the optimal stopping problem for the random walk X^{n} = {X_{k2^{−n}}: k ∈ N} discounted at rate α_{n} = e^{−δ2^{−n}} with reward g_{c}, (Theorem II.23 in Shiryaev (1978)).

Take now δ ≥ 0 and τ(δ) as in (4). For fixed n, and δ > 0

τ_{n} = \frac{[2^{n}τ(δ)] + 1}{2^{n}}

(with [x] the integer part of x) is a geometric random variable taking values on the set \{\frac{1}{2^{n}} , \frac{2}{2^{n}} , …, \frac{k}{2^{n}} , …\} with parameter α_{n} = e^{−δ2^{−n}}, i.e. P(τ_{n} > k2^{−n}) = (e^{−δ/2^{n}})^{k}. If δ = 0 put τ_{n} = ∞. On the other side Proposition 1 give the solution to this discrete time optimal stopping problem. In conclusion

lim_{N→∞} Q_{n}^{N}g_{c}(x) = E(x + M^{n} − EM^{n})^{+} (18)

where the r.h.s. is the formula in Proposition 1, and

M^{n} = sup{X_{k2^{−n}}: 0 ≤ k < τ_{n}},

M^{n} ≤ M and EM^{n} < ∞. Now we make n → ∞. First

(19) M_{n} = sup_{0≤k<τ_{n}} X_{k2^{−n}} = sup_{0≤k<τ(δ)} X_{k2^{−n}} → M = sup_{0≤t<τ(δ)} X_{t} P-a.s.

because τ_{n} > τ(δ), τ_{n} → τ(δ) and X is càdlàg. The result then follows taking limits in (18). The l.h.s. has limit V_{c}(x) by (17) and

E(x + M^{n} − EM^{n})^{+} → E(x + M − EM)^{+}

in view of (19) and dominated convergence since EM < ∞. The fact that τ_{c}^{∗} is the optimal stopping rule follows exactly as in the proof of (iii) in Proposition 1.

Proof of Theorem 2. We simply plug formula (11) in Theorem 1.

EM = ∫_{0}^{∞} yh_{n+1}(y, A, p)dy = ∑_{k=1}^{n+1} \frac{A_{k}}{p_{k}} .

Furthermore, if x ≥ x^{∗}, E(x + M − EM − K)^{+} = x − K. If x < x^{∗}

V_{c}(x) = E(x + M − EM − K)^{+} = ∫_{x^{∗}−x}^{∞}(x + y − x^{∗})h_{n+1}(y, A, p)dy

= ∑_{k=1}^{n+1} \frac{A_{k}}{p_{k}} e^{−p_{k}(x^{∗}−x)}.

concluding the proof.

Proof of Theorem 3. Let δ ≥ 0 and X be a Lévy process with EM finite. Assume x ≥ 0. As for any τ ∈ M inequality (x+X_{τ})^{+}1_{{τ<τ(δ)}} ≤ x+M, holds and Ee^{−δτ} (x + X_{τ})^{+} = E(x + X_{τ})^{+}1_{{τ<τ(δ)}}, we have

V (x) = sup_{τ} Ee^{−δτ} (x + X_{τ})^{+} ≤ E(x + M).

We have to prove now, that for x ≥ 0,

E(x + M) ≤ eV (x).

As V (x) = E(x + M − EM)^{+}, this amounts to say, that for x ≥ 0

q(x) = \frac{E(x + M)}{E(x + M − EM)^{+}} ≤ e. (20)

If x ≥ EM, E(x + M − EM)^{+} = x, and q(x) ≤ 2 < e.

Denoting B(x) = P(M ≥ x), we claim

B(x + y) ≥ B(x)B(y). (21)

To see this consider, for z ≥ 0

τ(z) = inf{t ≥ 0: X_{t} ≥ z}.

Based on X_{τ(x)} ≥ x on the set τ(x) < τ(δ), and the lack of memory of τ(δ)

B(x + y) = P( sup_{0≤t<τ(δ)} X_{t} ≥ x + y) = P( sup_{0≤t<τ(δ)} X_{t} ≥ x + y | τ(x) < τ(δ))B(x)

≤ P( sup_{0≤t<τ(δ)} X_{t} ≥ X_{τ(x)} + y | τ(x) < τ(δ))B(x)

= P( sup_{τ(x)≤t<τ(δ)} X_{t} − X_{τ(x)} ≥ y | τ(x) < τ(δ))B(x) = B(y)B(x),

and (21) is proved.

Consider now the auxiliary function f = f(x), given by

f(x) = e^{\frac{x}{x^{∗}}} ∫_{x}^{+∞} B(y)dy, x ≥ 0,

where x^{∗} = EM = ∫_{0}^{+∞} B(y)dy. Differentiation gives

f′(x) = e^{\frac{x}{x^{∗}}} (\frac{1}{x^{∗}} ∫_{x}^{+∞} B(x)dx − B(x))

In order to conclude that f′(x) ≥ 0 for all x ≥ 0, we integrate (21)

∫_{x}^{+∞} B(y)dy = ∫_{0}^{+∞} B(x + y)dy ≥ B(x) ∫_{0}^{+∞} B(y)dy = B(x)x^{∗}

and deduce that f(x) is non decreasing. As f(0) = x^{∗} we conclude

f(x) = e^{\frac{x}{x^{∗}}} ∫_{x}^{+∞} B(y)dy ≥ x^{∗}. (22)

Now, applying (22) with 0 ≤ x ≤ x^{∗}

q(x) = \frac{x + x^{∗}}{∫_{x^{∗}−x}^{+∞} B(y)dy} ≤ \frac{x + x^{∗}}{x^{∗}} exp (\frac{x^{∗} − x}{x^{∗}}) = c(x).

It is direct to see that the function c(x) is decreasing, and that c(0) = e, concluding the proof.

5 Conclusion

The presented paper shows how to reduce a discounted optimal stopping problem for a Lévy process X, with rewards (x − K)^{+} and (K − x)^{+} to the computation of the distribution of the supremum M and infimum I of the underlying process X killed at the discount rate. It is interesting to note, that in general, the optimal stopping problem of a Markov process, leds to a free boundary problem, and the computation of the supremum of a process has fixed boundary conditions. Second, recent results on exact distributions of M (resp. I) when positive (resp. negative) jumps of X are distributed according a mixture of exponentials and negative (resp. positive) are arbitrary are used, in order to obtain closed solutions to the optimal stopping problems. Third, a prophet type inequality is obtained, that can be seen as a comparison between prices of “perpetual look-back call options” and perpetual call options. The presented results complement the ones contained in the paper “Optimal Stopping and Perpetual Options for Lévy processes”, (Mordecki, (2000b)), where reward functions (e^{x} − K)^{+} and (K − e^{x})^{+} are considered, and applications to finance are discussed in detail.

References

[1] Bachelier L. Théorie de la spéculation. Ann. Ecole Norm. Sup., 1900, v. 17, pp 21–86. (Reprinted in: The random character of stock market prices. Ed. by P.H. Cootner Cambridge, Massachusetts: MIT Press, 1967. pp. 17–78.)

[2] Bertoin, J. (1996). Lévy Processes. Cambridge University Press, Cambridge.

[3] Darling, D.A., Ligget, T. Taylor, H.M. (1972). Optimal stopping for partial sums. The Annals of Mathematical Statistics, 43 1363–1368.

[4] Hill, T.P., Kertz, R.P. (1990). A survey of prophet inequalities in optimal stopping theory. Contemporary Mathematics, 125 Amer. Math. Soc., Providence, RI, 191–207.

[5] Jacod, J., Shiryaev, A.N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin Heidelberg.

[6] Kertz, R. Pricing-differentials and bounds for look-back options, and prophet problems in probability. Advances in stochastic inequalities (Atlanta, GA, 1997), Contemporary Mathematics, 234, Amer. Math. Soc., Providence, RI, 1999, 97–120.

[7] Mordecki, E. (1997). Ruin Probabilities and Optimal Stopping for a Diffusion with jumps. In Actas del IV Congreso Monteiro (Fernández Stacco E.L. et al. eds.) 39–48. Bahia Blanca, Argentina.

[8] Mordecki, E. (1998). Optimal stopping for a compound Poisson process with exponential jumps. Publicaciones Matemáticas del Uruguay, 7 55–66.

[9] Mordecki, E. (1999) Ruin probabilities for a Lévy process with mixed exponential negative jumps. Pre-Mat, 99/28. Submitted.

[10] Mordecki, E. (2000a) Optimal stopping and perpetual options for Lévy processes. Pre-Mat 2000/35. Submitted.

[11] Mordecki E., (2000b) Optimal stopping, ruin probabilities and prophet inequalities for Lévy processes. Pre-Mat 2000/38.

[12] Shiryaev, A.N. (1978). Optimal Stopping Rules. Springer, New York Heidelberg.

[13] Skorokhod, A. V. (1991). Random processes with independent increments. Kluwer Academic Publishers, Dordrecht.

[14] Taylor, H. M. (1968) Optimal stopping in a Markov process. The Annals of Mathematical Statistics, 39 1333–1344.

[15] Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999.

[16] Shiryaev, A. N. Essentials of stochastic finance. Facts, models, theory. Advanced Series on Statistical Science and Applied Probability, 3. World Scientific Publishing Co., Inc., River Edge, NJ, 1999.