Smoothing of paths and weak approximation of the occupation measure of Lévy processes

Ernesto Mordecki Mario Wschebor Centro de Matemática. Facultad de Ciencias. Universidad de la República. Calle Iguá 4225. 11400 Montevideo. Uruguay. mordecki@cmat.edu.uy, wschebor@cmat.edu.uy

September 10, 2003

Abstract Consider a real-valued Lévy process with non-zero Brownian component and jumps with locally finite variation. We obtain an invariance principle theorem for the speed of approximation of its occupation measure by means of functionals defined on regularizations of the paths.

AMS subject classification. Primary: 60G51, 60J75. Secondary: 60J55. Short Title: Regularized Lévy processes Key words and phrases: Lévy processes, occupation measure, smoothing of

paths.

1 Lévy processes

Let X = {X_{t}: t ≥ 0} be a real-valued Lévy process, defined on a probability space (Ω, F, P), that we represent by

X_{t} = σW_{t} + S_{t} + mt.  (1)

Here W = {W_{t}: t ≥ 0} is a standard Wiener process, S = {S_{t}: t ≥ 0} a pure jump process with càdlàg paths, m and σ are real constants, and we assume that the Gaussian part does not vanish, i.e. σ > 0. Denote by F = {F_{t}: t ≥ 0} the minimal filtration generated by X, that satisfy the usual assumptions (see Jacod and Shiryaev (1987)).

Furthermore, assume that

(FV) the jump part of the process has locally finite variation, i.e. for each positive t, ∑_{0<r≤t} |ΔS_{r}| is almost surely finite, where, as usual, we denote f(r−) the left limit of a càdlàg function f on the point r, and Δf_{r} = f(r) − f(r−) is the magnitude of its jump at this point.

In view of (FV), the random variable S_{t} satisfies

S_{t} = ∑_{0<s≤t} ΔX_{s}.

Given a positive constant a, it will be useful to define the processes S^{a} = {S_{t}^{a}: t ≥ 0} and X^{a} = {X_{t}^{a}: t ≥ 0} by

S_{t}^{a} = ∑_{0<s≤t} ΔX_{s}1_{{|ΔX_{s}|≥a}},  (2)

that is the (a.s. finite) sum of jumps of the process greater or equal than a, and

X_{t}^{a} = mt + σW_{t} + S_{t}^{a},  (3)

respectively.

The characteristic function of the random variable X_{t} has the standard form E (e^{zX_{t}}) = e^{tκ(z)}, where the function κ(z) (defined for the complex values of z such that this expectation is finite) has the form

κ(z) = mz + \frac{1}{2}σ^{2}z^{2} + ∫_{ℝ} (e^{zy} − 1)Π(dy).  (4)

Here Π(dy), the Lévy-Khinchine measure of the process, is a non-negative measure defined on ℝ \ {0} that, in accordance with condition (FV) above, satisfies ∫ (1 ∧ |y|)Π(dy) < ∞. We denote by ν_{t}(dy) the Poisson jump measure of the process on the interval [0, t]. Note that for each t > 0, we have a.s. ν_{t}({|x| ≥ δ}) < ∞ for every δ > 0. For general references on Lévy processes see Skorokhod (1991), Bertoin (1996) or Sato (1999).

2 Regularized Lévy processes

We now describe the regularization of the trajectories, that, in our context, is interpreted as a partial observation of the process through a physical device. Let ψ: ℝ → ℝ^{+} be a C^{1} function with compact support, say supp(ψ) ⊂ [−1, 1], such that ∫_{−1}^{1} ψ(t)dt = 1 and, for ε > 0, define the approximation of unity

ψ_{ε}(t) = \frac{1}{ε}ψ(\frac{t}{ε}).

We denote by ∥ψ∥ = ( ∫_{−1}^{1} ψ^{2}(t)dt)^{1/2} the norm of ψ in L^{2}(ℝ, dt). The regularization X^{ε} = {X_{t}^{ε}: t ≥ 0} of the process is obtained by convolution with ψ_{ε} in the following way:

X_{t}^{ε} = (ψ_{ε} ∗ X)_{t} = ∫_{ℝ} ψ_{ε}(t − s)X_{s}ds = ∫_{−1}^{1} ψ(−w)X_{t+wε}dw,  (5)

where we set X_{s} = W_{s} = S_{s} = 0 if s < 0. In the same way, we define W^{ε} = {W_{t}^{ε}: t ≥ 0} and S^{ε} = {S_{t}^{ε}: t ≥ 0}, and obtain that X_{t}^{ε} = mt−εmα+σW_{t}^{ε} +S_{t}^{ε} where α = ∫_{ℝ} wψ(w)dw.

Observe that the regularized processes inherits the regularity properties of ψ, so that X^{ε} has C^{1} paths. For further reference, we compute the timederivative (denoted with a dot) of the regularized process, that can be written as a stochastic integral:

Ẋ_{t}^{ε} = ∫_{ℝ} \frac{∂}{∂t}(ψ_{ε}(t − s))X_{s}ds = \frac{1}{ε} ∫_{−1}^{1} ψ̇(−w) (X_{t+εw} − X_{t−ε}) dw
= ∫_{ℝ} ψ_{ε}(t − s)dX_{s} = \frac{1}{ε} ∫_{−1}^{1} ψ(−w)d^{w}(X_{t+εw}).  (6)

Similar formulae hold for W^{ε}, S^{ε} and S^{a,ε}. In particular,

Ẇ_{t}^{ε} = \frac{1}{ε} ∫_{−1}^{1} ψ̇(−w)(W_{t+εw} − W_{t−ε})dw,

which implies, for t ∈ [0, T] and 0 < ε < 1:

|εẆ_{t}^{ε}| ≤ 2∥ψ̇∥_{∞} sup_{|h|<ε,t∈[0,T]} |W_{t+h} − W_{t−ε}| ≤ C_{η}(ω)ε^{1/2−η},  (7)

with η ∈ (0, 1/2) arbitrary, and C_{η}(ω) a random constant independent of ε; we also have that \sqrt{ε}Ẇ_{t}^{ε} has centered Gaussian distribution, with variance ∥ψ∥^{2}.

If F: ℝ^{+} → ℝ is a C^{1} function, we denote the number of crossings of the level u by the function F on an interval I = [s, t], by

N_{u}^{F} [s, t] = ♯{r: F_{r} = u, r ∈ I},  (8)

that is, the number of roots belonging to I of the equation F_{t} = u. It is easy to verify, that, for a given continuous function f: ℝ → ℝ, we have

∫_{−∞}^{∞} f(u)N_{u}^{F} [0, T]du = ∫_{0}^{T} f(F_{t})|Ḟ_{t}|dt.  (9)

3 Main Result

The aim of Theorem 1 below is to approximate the occupation measure of the process X on the interval [0, T] by a re-normalization of the number of crossings of the process X^{ε} = {X_{t}^{ε}: t ≥ 0} with horizontal levels on the same time interval.

Theorem 1 Consider a Lévy process X = {X_{t}: t ≥ 0} with characteristic exponent given in (4), σ > 0, finite variation jump component, and the regularization X^{ε} = {X_{t}^{ε}: t ≥ 0} defined in (5). Then, for each C^{2}-function f: ℝ → ℝ with bounded second derivative, we have

\frac{1}{\sqrt{ε}} [∫_{ℝ} f(u)C_{ε}\sqrt{ε}N_{u}^{X^{ε}}[0, t]du − σ ∫_{0}^{t} f(X_{s})ds] − C_{ε} ∫_{0}^{t} f(X_{s}^{ε})|Ṡ_{s}^{ε}|ds

⇒ D ∫_{0}^{t} f(X_{s})dB_{s} (10)

as ε → 0, where:

• B = {B_{t}: t ≥ 0} is a Wiener process independent of X;

• The first constant is

C_{ε} = \frac{σ}{E(\sqrt{ε}|σẆ_{1}^{ε} + m|)} → \frac{1}{∥ψ∥}\sqrt{\frac{π}{2}} = C_{0} (ε → 0).  (11)

• The second constant is

D^{2} = 2σ^{2} ∫_{0}^{2} (r(t) Arsin r(t) + \sqrt{1 − r^{2}(t)} − 1)dt,  (12)

where r(t) is a covariance function defined by

r(t) = \frac{1}{∥ψ∥^{2}} ∫ ψ(t − u)ψ(−u)du.

• ⇒ denotes weak convergence in the space C = C([0, +∞) , ℝ) of continous functions.

Before proving the Theorem we make some remarks on the statement.

Remarks.

1.- A simple consequence of Theorem 1 is that for each t > 0, one has

∫_{ℝ} f(u)C_{ε}\sqrt{ε}N_{u}^{X^{ε}}[0, t]du → σ ∫_{0}^{t} f(X_{s})ds in probability  (13)

as ε → 0. This result can be used to estimate σ from the observation of the smoothed path X^{ε}. Results of type (13) are well-known for semimartingales having continuous paths (Azaïs & Wschebor, 1997) and also other classes of processes (Azaïs & Wschebor, 1996), where almost sure convergence is proved.

2.- Theorem 1 contains the speed of convergence in (13). This allows to make inference on σ from the observation of X^{ε}.

Analogous results for processes with continuous paths are in Berzin & León (1994) for Brownian motion and in Perera & Wschebor (1998, 2002) for certain classes of continuous semi-martingales having Itô-integrals as martingale part. Even if X is a Brownian motion, the proof below seems to be simpler and more direct than previously published ones.

There exist also some related results for Brownian motion and general diffusions, where the approximation X^{ε} of the actual path X is replaced by polygonal approximation and the smooth function f by a Dirac-delta function, or considering functionals defined on random walks. See for example, Dacunha-Castelle and Florens (1986), Florens (1993), Génon-Catalot and Jacod (1993), Borodin and Ibragimov (1994), and Jacod (1998, 2000). In this context, if ε is the size of the discretization in time, then the speed of convergence turns out to be of the order ε^{1/4}.

3.- We shall prove (see Proposition 3 in next section) that for each t > 0 the bias term

L_{ε}(f, t) = C_{ε} ∫_{0}^{t} f(X_{s}^{ε})|Ṡ_{s}^{ε}|ds  (14)

in (10), almost surely converges, as ε → 0, to

L_{0}(f, t) = C_{0} ∑_{0<s≤t} L(f, s)|ΔX_{s}|  (15)

where

L(f, t) = ∫_{−1}^{1} ψ(z)f(X_{t−} ∫_{z}^{1} ψ(w)dw + X_{t} ∫_{−1}^{z} ψ(w)dw)dz.  (16)

It follows that one can replace Theorem 1 by the statement

\frac{1}{\sqrt{ε}} [∫_{ℝ} f(u)C_{ε}\sqrt{ε}N_{u}^{X^{ε}}[0, t]du − σ ∫_{0}^{t} f(X_{s})ds] − L_{0}(f, t)  (17)

converges cilindrically to the law of D ∫_{0}^{t} f(X_{s})dB_{s}.

With this statement, the bias term does not depend on ε, but excepting the case of trivial f, we lose weak convergence when the jump part of X does not vanish.

4 Proofs

Proof of Theorem 1

In order to prove the Theorem we first observe, in view of (9), that

∫_{ℝ} f(u)N_{u}^{X^{ε}}[0, t]du = ∫_{0}^{t} f(X_{s}^{ε})|Ẋ_{s}^{ε}|ds, a.s.

Write our expression as the sum of three terms:

\frac{1}{\sqrt{ε}} ∫_{0}^{t} f(X_{s}^{ε})C_{ε}\sqrt{ε}|Ẋ_{s}^{ε}|ds − \frac{σ}{\sqrt{ε}} ∫_{0}^{t} f(X_{s})ds − C_{ε} ∫_{0}^{t} f(X_{s}^{ε})|Ṡ_{s}^{ε}|ds

= C_{ε} ∫_{0}^{t} f(X_{s}^{ε})(|Ẋ_{s}^{ε}| − |σẆ_{s}^{ε} + m| − |Ṡ_{s}^{ε}|)ds

+ C_{ε} ∫_{0}^{t} f(X_{s}^{ε})|σẆ_{s}^{ε} + m|ds − C_{ε} ∫_{0}^{t} f(X_{s})|σẆ_{s−ε}^{ε} + m|ds

+ \frac{1}{\sqrt{ε}} ∫_{0}^{t} (C_{ε}\sqrt{ε}|σẆ_{s−ε}^{ε} + m| − σ)f(X_{s})ds.

We now introduce the following simplification, that will be useful for the proof. Given an arbitrary δ ∈ (0, 1) there exists b > 0 such that there is no jump with absolute value greater than b, with probability greater that 1 − δ. The given Lévy process can be written as X_{t} = (X_{t} − S_{t}^{b}) + S_{t}^{b}, where the random processes {S_{t}^{b} : t ≥ 0} (defined in (2)) and {X_{t} − S_{t}^{b} : t ≥ 0} are independent. A standard argument shows that it is enough to prove the result for the process {X_{t} − S_{t}^{b} : 0 ≤ t ≤ T }, so, in what follows, we assume that the support of N is contained in the interval [−b, b]. Under this additional hypothesis, it is easy to see that for each t ≥ 0 the random variable X_{t} has finite moments of all orders.

In what follows, the parameter of the various processes we will consider vary in a fixed interval [0, T].

We divide the proof into three steps:

1. Proof of

Z_{t}^{1,ε} = ∫_{0}^{t} f(X_{s}^{ε})(|Ẋ_{s}^{ε}| − |σẆ_{s}^{ε} + m| − |Ṡ_{s}^{ε}|)ds ⇒ 0.  (18)

2. Proof of

Z_{t}^{2,ε} = ∫_{0}^{t} f(X_{s}^{ε})|σẆ_{s}^{ε} + m|ds − ∫_{0}^{t} f(X_{s})|σẆ_{s−ε}^{ε} + m|ds ⇒ 0 (19)

3. Proof of

Z_{t}^{3,ε} = \frac{1}{\sqrt{ε}} ∫_{0}^{t} (C_{ε}\sqrt{ε}|σẆ_{s−ε}^{ε} + m| − σ)f(X_{s})ds ⇒ D ∫_{0}^{t} f(X_{t})dB_{s}. (20)

Proof of Step 1. We prove

sup_{0≤t≤T} | ∫_{0}^{t} f(X_{s}^{ε})(|Ẋ_{s}^{ε}| − |σẆ_{s}^{ε} + m| − |Ṡ_{s}^{ε}|)ds| → 0 a.s. (ε → 0).

Since there exists an almost surely finite random variable M(ω) such that

sup_{0≤s≤T} |f(X_{s}^{ε})| ≤ M(ω),  sup_{0≤s≤T} |f(X_{s})| ≤ M(ω),  (21)

it is enough to prove that

∫_{0}^{T} ||Ẋ_{s}^{ε}| − |σẆ_{s}^{ε} + m| − |Ṡ_{s}^{ε}||ds → 0 a.s. (ε → 0).

We have

||Ẋ_{s}^{ε}| − |σẆ_{s}^{ε} + m| − |Ṡ_{s}^{ε}|| ≤ 2 min (|σẆ_{s}^{ε} + m|, |Ṡ_{s}^{ε}|)
≤ 2 min (|σẆ_{s}^{ε} + m|, |Ṡ_{s}^{a,ε}|) + 2|Ṡ_{s}^{a,ε} − Ṡ_{s}^{ε}|.

We claim that

sup_{0<ε≤1} ∫_{0}^{T} |Ṡ_{s}^{ε} − Ṡ_{s}^{a,ε}|ds → 0 a.s. (a → 0).  (22)

In fact, denoting g(x) = |x|1_{{|x|<a}}, in view of (6), we have

|Ṡ_{t}^{ε} − Ṡ_{t}^{a,ε}| ≤ |\frac{1}{ε} ∫_{−1}^{1} |ψ̇(−w)| ∑_{t−ε<v≤t+εw} g(ΔX_{v})dw|

≤ \frac{2}{ε}∥ψ̇∥_{∞} ∑_{t−ε<v≤t+ε} g(ΔX_{v}).

Furthermore, if G(t) = ∑_{0<v≤t} g(ΔX_{v}) and 0 < ε ≤ 1, we obtain

∫_{0}^{T} |Ṡ_{s}^{ε} − Ṡ_{s}^{a,ε}|ds ≤ \frac{2}{ε}∥ψ̇∥_{∞} ∫_{0}^{T} (G(t + ε) − G(t − ε))dt

= \frac{2}{ε}∥ψ̇∥_{∞} ∫_{0}^{T} ( ∫_{t−ε}^{t+ε} G(dv))dt

≤ \frac{2}{ε}∥ψ̇∥_{∞} ∫_{0}^{T +1} ( ∫_{v−ε}^{v+ε} dt)G(dv)

= 4∥ψ̇∥_{∞}G(T + 1)

= 4∥ψ̇∥_{∞} ∑_{0<t≤T +1} |ΔX_{t}|1_{{|ΔX_{t}|<a}},

and (22) follows.

Consider now a fixed. Put τ_{0} = 0, and denote the succesive epochs of jump with absolute value not smaller that a by

τ_{n} = inf{t > τ_{n−1} : |ΔX_{t}| ≥ a} (n = 1, 2, …).

Denote also by N_{t} = max{n: τ_{n} ≤ t} (t ≥ 0) the number of these jumps up to time t. Fix ω ∈ Ω, and choose ε > 0 such that the intervals (τ_{n} − ε, τ_{n} + ε) (n = 1, …, N_{T}) are disjoint. Taking into account that Ṡ^{a,ε} = 0 outside these intervals, and applying (7) with η = 1/4, we obtain for s ∈ [0, T]:

min (|σẆ_{s}^{ε} + m|, |Ṡ_{s}^{a,ε}|) ≤ Ĉ_{1/4}(ω)ε^{−3/4} ∑_{n=1}^{N_{T}} 1_{(τ_{n}−ε,τ_{n}+ε)}(s).

where Ĉ_{1/4} is a new constant depending on ω ∈ Ω, and on the parameters m, σ. We then obtain

∫_{0}^{T} min (|σẆ_{s}^{ε} + m|, |Ṡ_{s}^{a,ε}|)ds ≤ ∑_{n=1}^{N_{T}} ∫_{τ_{n}−ε}^{τ_{n}+ε} Ĉ_{1/4}(ω)ε^{−3/4}ds
= 2Ĉ_{1/4}(ω)N_{T} (ω)ε^{1/4}.

From this inequality and (22) the statement of Step 1 follows.

Proof of Step 2. First observe that

∫_{0}^{t} f(X_{s})|σẆ_{s−ε}^{ε} + m|ds = ∫_{−ε}^{t−ε} f(X_{s+ε})|σẆ_{s}^{ε} + m|ds

which implies

Z_{t}^{2,ε} = ∫_{0}^{t} (f(X_{s}^{ε}) − f(X_{s+ε}))|σẆ_{s}^{ε} + m|ds
− ∫_{−ε}^{0} f(X_{s+ε})|σẆ_{s}^{ε} + m|ds + ∫_{t−ε}^{t} f(X_{s+ε})|σẆ_{s}^{ε} + m|ds.

For the second integral, we have

| ∫_{−ε}^{0} f(X_{s+ε})|σẆ_{s}^{ε} + m|ds| ≤ M(ω)Ĉ_{1/4}(ω)ε^{1/4},

where M(ω) is given in (21). (Remember that W_{s} = 0 if s < 0, but W_{s}^{ε} does not necessarily vanishes for s < 0.) A similar bound holds for the third integral.

So, in order to obtain (19), we must prove that

Ẑ_{t}^{2,ε} = ∫_{0}^{t} (f(X_{s}^{ε}) − f(X_{s+ε}))|σẆ_{s}^{ε} + m|ds ⇒ 0  (23)

Denote

f_{t}^{ε} = f(X_{t}^{ε}) − f(X_{t+ε}), g_{t}^{ε} = |σẆ_{t}^{ε} + m|.

Given S < T we compute the second moment:

E(Ẑ_{T}^{2,ε} − Ẑ_{S}^{2,ε})^{2} = 2 ∫∫ _{S≤s+2ε≤t≤T} E(f_{s}^{ε}f_{t}^{ε}g_{s}^{ε}g_{t}^{ε})dsdt  (24)

+ 2 ∫∫ _{S≤s≤t≤s+2ε≤T} E(f_{s}^{ε}f_{t}^{ε}g_{s}^{ε}g_{t}^{ε})dsdt = 2(L_{1} + L_{2}).

We denote by Δ_{t}^{ε} the increment

Δ_{t}^{ε} = X_{t}^{ε} − X_{t+ε} = ∫_{−1}^{1} ψ(−w)(X_{t+wε} − X_{t+ε})dw
= σΔ_{t}^{ε,W} + Δ_{t}^{ε,S} − εm(α + 1)

(with the obvious notation). Furthermore

E(Δ_{t}^{ε,W} )^{2} = ∫∫ ψ(−u)ψ(−v)E((W_{t+εu} − W_{t+ε})(W_{t+εv} − W_{t+ε}))dudv
= ε ∫∫ ψ(−u)ψ(−v)(1 − u ∨ v)dudv.

Now we apply Taylor’s expansion:

f_{t}^{ε} = f(X_{t+ε} + Δ_{t}^{ε}) − f(X_{t+ε}) = f′(X_{t+ε})Δ_{t}^{ε} + \frac{1}{2}f′′(X_{t+ε} + θΔ_{t}^{ε})(Δ_{t}^{ε})^{2}/2
= f′(X_{t−ε})Δ_{t}^{ε} + (f′(X_{t+ε}) − f′(X_{t−ε}))Δ_{t}^{ε}  (25)
+ \frac{1}{2}f′′(X_{t+ε} + θΔ_{t}^{ε})(Δ_{t}^{ε})^{2}/2  (26)

where 0 < θ < 1.

Take now conditional expectations in the integrand corresponding to L_{1} in (28):

E(f_{s}^{ε}f_{t}^{ε}g_{s}^{ε}g_{t}^{ε}) = E(f_{s}^{ε}g_{s}^{ε}E(f_{t}^{ε}g_{t}^{ε}/F_{t−ε})).

Plug the Taylor expansion for f_{t}^{ε} into the last expectation and consider each term. First, as Δ_{t}^{ε}g_{t}^{ε} is independent of F_{t−ε},

E(f′(X_{t−ε})Δ_{t}^{ε}g_{t}^{ε}/F_{t−ε}) = f′(X_{t−ε})E(Δ_{t}^{ε}g_{t}^{ε})

= f′(X_{t−ε}) [σE(Δ_{t}^{ε,W} g_{t}^{ε}) + E(Δ_{t}^{ε,S})E(g_{t}^{ε}) − εm(α + 1)E(g_{t}^{ε})]

since S and W are independent processes. For the first term in brackets, substracting E(Δ_{t}^{ε,W} |σẆ_{t}^{ε}|) = 0, we have

|E(Δ_{t}^{ε,W} g_{t}^{ε})| = |E(Δ_{t}^{ε,W} (|σẆ_{t}^{ε} + m| − |σẆ_{t}^{ε}|))|
≤ |m|E|Δ_{t}^{ε,W} | = (const)ε^{1/2}.

In what concerns the second term in brackets,

E|Δ_{t}^{ε,S}| ≤ ∥ψ∥_{∞}E( ∑_{t−ε<s≤t+ε} |ΔX_{s}|) = ∥ψ∥_{∞}2ε ∫ |x|Π(dx),

so that

|E(Δ_{t}^{ε,S})E(g_{t}^{ε})| ≤ (const)ε^{1/2},

and we obtain:

|E(f′(X_{t−ε})Δ_{t}^{ε}g_{t}^{ε} | F_{t−ε})| ≤ |f′(X_{t−ε})| (const) ε^{1/2}

≤ (∥f′′∥_{∞}|X_{t−ε}| + 1)(const)ε^{1/2}.

Furthermore

|E((f′(X_{t+ε}) − f′(X_{t−ε}))Δ_{t}^{ε}g_{t}^{ε} | F_{t−ε})|

= |E((f′′(X_{t−ε} + θ′(X_{t+ε} − X_{t−ε}))(X_{t+ε} − X_{t−ε})Δ_{t}^{ε}g_{t}^{ε} | F_{t−ε})|

≤ ∥f′′∥_{∞}E(|σ(W_{t+ε} − W_{t−ε}) + S_{t+ε} − S_{t−ε} + 2mε| × |Δ_{t}^{ε}|g_{t}^{ε}),

where 0 < θ′ < 1. A standard computation with normal distributions shows that:

E(|Δ_{t}^{ε,W} |^{2} [g_{t}^{ε}]^{2} ) ≤ (const)

So, by Cauchy-Schwarz’s inequality we obtain

E(|σ(W_{t+ε} − W_{t−ε}) + S_{t+ε} − S_{t−ε} + 2mε| × |Δ_{t}^{ε,W} |g_{t}^{ε}) ≤ (const)ε^{1/2}. Also

E(|W_{t+ε} − W_{t−ε}| × |Δ_{t}^{ε,S}|g_{t}^{ε}) = E(|W_{t+ε} − W_{t−ε}|g_{t}^{ε})E|Δ_{t}^{ε,S}| ≤ (const)ε.

As for the other term

E(|S_{t+ε} − S_{t−ε}| × |Δ_{t}^{ε,S}|g_{t}^{ε})

= E(|(S_{t+ε} − S_{t−ε})Δ_{t}^{ε,S}|)E|g_{t}^{ε}| ≤ (const)ε^{1/2},

because E|g_{t}^{ε}| ≤ (const)ε^{−1/2} and

E(|(S_{t+ε} − S_{t−ε})Δ_{t}^{ε,S}|)

= E| ∫_{−1}^{1} ψ(−w)(S_{t+ε} − S_{t−ε})(S_{t+wε} − S_{t+ε})dw| ≤ (const)ε.

Let us now consider the result of plugging the last term of (26) into the conditional expectation. We have:

|E(f′′(X_{t+ε} + θΔ_{t}^{ε})(Δ_{t}^{ε})^{2}g_{t}^{ε}/F_{t−ε})| ≤

(const)∥f′′∥_{∞}E(σ^{2}(Δ_{t}^{ε,W})^{2}g_{t}^{ε} + (Δ_{t}^{ε,S})^{2}g_{t}^{ε} + ε^{2}g_{t}^{ε}) ≤ (const)ε^{1/2}, (27) based on similar computations. Summing up, we obtain (in the integral L_{1}):

E(f_{s}^{ε}g_{s}^{ε}/F_{t−ε}) ≤ (const)ε^{1/2}.

This also shows that

E(f_{s}^{ε}g_{s}^{ε}) ≤ (const)ε^{1/2},

so that

L_{1} ≤ (const)(T − S)^{2}ε.  (28)

On the other hand, let us show that for s, t ∈ [0, T] and 0 < ε ≤ 1 the expectation E(f_{s}^{ε}f_{t}^{ε}g_{s}^{ε}g_{t}^{ε}) is bounded.

Applying Cauchy-Schwarz’s inequality, it suffices to prove the boundedness of

E {(f_{t}^{ε}g_{t}^{ε})^{2}}

for t ∈ [0, T] and 0 < ε < 1. Check that

E {(f_{t}^{ε}g_{t}^{ε})^{2}} ≤ (const) (∥f′′∥_{∞} + 1)^{2} ×

[E {X_{t−ε}^{2} (Δ_{t}^{ε}g_{t}^{ε})^{2} + E {(Δ_{t}^{ε})^{4} (g_{t}^{ε})^{2}}} + E {(X_{t+ε} − X_{t−ε})^{2}(Δ_{t}^{ε})^{2}(g_{t}^{ε})^{2}}] ,

and the proof of the boundedness of this expression follows in much a similar way as the one of L_{1}.

This implies, first, that

E((Ẑ_{T}^{2,ε} − Ẑ_{S}^{2,ε})^{2}) ≤ (const)(T − S)^{2}

for 0 ≤ S, T ≤ T_{0}, hence that {Ẑ_{T}^{2,ε}: 0 ≤ T ≤ T_{0}} is tight in C([0, T_{0}], ℝ) and, second, that

E((Ẑ_{T}^{2,ε})^{2}) ≤ (const)T^{2}ε,

so that, for fixed T, Ẑ_{T}^{2,ε} → 0 (ε → 0) in L^{2}. This proves (23).

Proof of Step 3. Introduce the processes y^{ε} = {y_{t}^{ε}: t ≥ 0} and Y^{ε} = {Y_{t}^{ε}: t ≥ 0} defined by

y_{t}^{ε} = C_{ε}\sqrt{ε}|σẆ_{t−ε}^{ε} + m| − σ,  Y_{t}^{ε} = \frac{1}{\sqrt{ε}} ∫_{0}^{t} y_{s}^{ε}ds,  t ≥ 0.

Let us prove that

Y^{ε} ⇒ DB,  (29)

where B = {B_{t}: t ≥ 0} is a Wiener process independent of X, and D the constant in (12).

In order to see this, first observe that, since y_{t}^{ε} depends on the increments of the process W on the interval [t − ε, t + ε], the process y^{ε} is 2ε-dependent, and as a consequence, the process Y^{ε} has asymptotically independent increments as ε → 0. This means that any cluster point in the weak topology for the family of processes {Y^{ε}: t ≥ 0} as ε → 0 is a process with independent increments. Analogous arguments give that any cluster point has stationary increments. In order to complete the proof of (29), we prove the tightness in the space of continuous functions.

E(Y_{t}^{ε} − Y_{s}^{ε})^{4} = \frac{1}{ε^{2}} E( ∫_{s}^{t} y_{u}^{ε}du)^{4}

= \frac{4!}{ε^{2}} ∫_{s}^{t} du_{1} ∫_{u_{1}}^{u_{1}+2ε} du_{2} ∫_{u_{2}}^{t} du_{3} ∫_{u_{3}}^{u_{3}+2ε} du_{4}E(y_{u_{1}}^{ε}y_{u_{2}}^{ε}y_{u_{3}}^{ε}y_{u_{4}}^{ε})

≤ 4 × 4!(s − t)^{2}E(y_{u}^{ε})^{4} ≤ (const)(t − s)^{2},

where we have used (i) the 2ε-dependence of {Ẇ_{t}^{ε}: t ≥ 0}, (ii) the fact that due to the choice of C_{ε} we have E(y_{t}^{ε}) = 0, and (iii) the fact that E(y_{u}^{ε})^{4} converges to a finite limit, as ε → 0. This proves the tightness property (see 12.51 in Billingsley (1968)). As Y^{ε} is a centered process, in order to conclude (29) it remains to compute the constant D. This constant can be obtained as

D^{2} = lim_{ε→0} E(Y_{1}^{ε})^{2}.

Now,

E(Y_{1}^{ε})^{2} = \frac{1}{ε} ∫_{0}^{1} ∫_{0}^{1} E(y_{s}^{ε}y_{t}^{ε})dsdt = \frac{2}{ε} ∫_{0}^{1} dt ∫_{t}^{(t+ε)∧1} E(y_{s}^{ε}y_{t}^{ε})

∼ \frac{2}{ε} ∫_{0}^{2ε} E(y_{1}^{ε}y_{1+t}^{ε})dt = 2 ∫_{0}^{2} E(y_{1}^{ε}y_{1+εu}^{ε})dt → 2σ^{2} ∫_{0}^{2} E(g(U_{0})g(U_{u}))du.

with U defined in (30), and g(x) defined in (32). The rest of the computation of the constant D is presented in the following result.

Given ε > 0 define the process U^{ε} = {U_{t}^{ε}: t ≥ 0} by

U_{t}^{ε} = \frac{\sqrt{ε}}{∥ψ∥}Ẇ_{εt−ε}^{ε}  (30)

For t ≥ 2, U^{ε} is a centered Gaussian stationary process with covariance function

r(t) = E(U_{2}^{ε}U_{2+t}^{ε}) = \frac{ε}{∥ψ∥^{2}} E(Ẇ_{ε}^{ε}Ẇ_{ε(1+t)}^{ε})
= \frac{1}{∥ψ∥^{2}} ∫ ψ(t − u)ψ(−u)du,  (31)

(where we used (6)). We conclude that the distribution of U^{ε} does not depend on ε (excluding the interval [0, 2]), and introduce the process U as a centered Gaussian stationary process with covariance given by (31), that can be put in place of U^{ε} for our purposes. Observe that E(U_{t})^{2} = 1.

Lemma 2 Define

g(x) = \sqrt{\frac{π}{2}} |x| − 1, (x ∈ ℝ).  (32)

Then (1) E(g(U_{t})) = 0.

(2) E(g(U_{0})g(U_{t})) = r(t) Arsin r(t) + \sqrt{1 − r^{2}(t)} − 1.

Proof. As U_{0} is a standard Gaussian random variable (1) is direct. In order to see (2), denote by

p(x, y, r) = \frac{1}{2π\sqrt{1 − r^{2}}} exp \{\frac{−1}{2(1 − r^{2})}[x^{2} + y^{2} − 2rxy]\},

the density of the Gaussian bidimensional vector (U_{0}, U_{t}) with r = r(t). If we denote f(r) = E(g(U_{0})g(U_{t})), it is not difficult to verify the following formal calculations:

f′′(r) = \frac{∂}{∂r} ∫∫ _{ℝ^{2}} g(x)g(y)\frac{∂}{∂r} p(x, y, r)dxdy

= \frac{∂}{∂r} ∫∫ _{ℝ^{2}} g(x)g(y)\frac{∂^{2}}{∂x∂y} p(x, y, r)dxdy

= \frac{∂}{∂r} ∫∫ _{ℝ^{2}} g′(x)g′(y)p(x, y, r)dxdy = \frac{∂}{∂r} f′(r)  (33)

= ∫∫ _{ℝ^{2}} g′′(x)g′′(y)p(x, y, r)dxdy = 2πp(0, 0, r) = \frac{1}{\sqrt{1 − r^{2}}} .

Here g′′(x) = 2\sqrt{π/2}δ_{0} where δ_{0} denotes a Dirac delta function at the origin, we twice use \frac{∂}{∂r} p(x, y, r) = \frac{∂^{2}}{∂x∂y} p(x, y, r), and twice integrate by parts. When r = 0 the random variables U_{0} and U_{t} are independent. This gives f(0) = E(g(U_{0})g(U_{t})) = E(g(U_{0}))^{2} = 0, by (1); and, by the intermediate step (33) we also have f′(0) = E(g′(U_{0})g′(U_{t})) = E(g′(U_{0}))^{2} = 0. Finally, integrating twice we get

E(g(U_{0})g(U_{t})) = r(t) Arsin r(t) + \sqrt{1 − r^{2}(t)} − 1,

concluding the proof of the Lemma.

We now claim that

(Y^{ε}, W) ⇒ (DB, W)  (34)

where (B, W) is a pair of standard independent Wiener processes.

For this, see first that

E((C_{ε}\sqrt{ε}|σẆ_{t−ε}^{ε} + m| − σ)W_{s})

= E((C_{ε}\sqrt{ε}|σẆ_{t−ε}^{ε} + m| − σ)(W_{t∧s} − W_{(t−ε)∧s})).

Now

|E(Y_{t}^{ε}W_{s})| = |\frac{1}{D\sqrt{ε}} ∫_{0}^{t} E((C_{ε}\sqrt{ε}|σẆ_{r−ε}^{ε} + m| − σ)W_{s})dr|

= |\frac{1}{D\sqrt{ε}} ∫_{(s−ε)∧t}^{s∧t} E((C_{ε}\sqrt{ε}|σẆ_{r}^{ε} + m| − σ)W_{s})dr|

≤ \frac{ε}{D\sqrt{ε}}(E(C_{ε}\sqrt{ε}|σẆ_{t−ε}^{ε} + m| − σ)^{2}E(W_{t} − W_{t−ε})^{2})^{1/2}

= (const)ε.

This means that E(Y_{t}^{ε}W_{s}) → 0 (ε → 0), and, as it is direct to obtain that {Y_{t}^{ε}W_{s}}_{ε>0} is uniformly integrable, we obtain (34). As a consequence, since the jump part is independent from the continous part in our Lévy process, we have the weak convergence

(Y^{ε}, W, S) ⇒ (DB, W, S)

where B is independent of X.

Let us finally see (20). Observe that for each ε > 0, a.s. the process Y^{ε} has locally finite variation. Applying Ito’s formula:

\frac{1}{\sqrt{ε}} ∫_{0}^{T} (C_{ε}\sqrt{ε}|σẆ_{t}^{ε} + m| − σ)f(X_{t})dt = ∫_{0}^{T} f(X_{t})dY_{t}^{ε}
= f(X_{T})Y_{T}^{ε} − ∫_{0}^{T} Y_{t}^{ε}df(X_{t}), (35)

where using the hypothesis that f is C^{2} it follows that {f(X_{t})} is a semimartingale. The process (Y^{ε}, X) is adapted, and weakly converges to (B, X).

As the integrator in the right hand member of (35) is fixed one can verify that the hypotheses of Theorem 2.2 in Kurtz and Protter (1991, see Remark 2.5) hold true, thus obtaining

f(X_{T})Y_{T}^{ε} − ∫_{0}^{T} Y_{t}^{ε}df(X_{t}) ⇒ f(X_{T})B_{T} − ∫_{0}^{T} B_{t}df(X_{t})

Now, we apply Ito’s formula, taking into account that the quadratic covariation [X, B] = 0 and we get

f(X_{T})B_{T} − ∫_{0}^{T} B_{t}df(X_{t}) = ∫_{0}^{T} f(X_{t})dB_{t},

completing the proof of (20).

To finish, we state and prove the proposition announced in Remark 3 after the statement of Theorem 1.

Proposition 3 Assume that X = {X_{t}: t ≥ 0} and f satisfy the hypothesis of Theorem 1. Then, for the processes defined in (15) and (14), for each t ≥ 0, almost surely

L_{ε}(f, t) → L_{0}(f, t),

as ε → 0.

Proof. On account of (22) it suffices to show that for fixed a > 0, almost surely

C_{0} ∫_{0}^{t} f(X_{s}^{a,ε})|Ṡ_{s}^{a,ε}|ds → L_{0}^{a}(f, t),  (36)

as ε → 0, where L_{0}^{a}(f, t) is obtained from (15) when the process X is replaced by X^{a}.

Using the same notations as in the last part of Proof of Step 1 in Theorem 1, we can write a.s., for ε sufficiently small

∫_{0}^{t} f(X_{s}^{a,ε})|Ṡ_{s}^{a,ε}|ds = ∑_{n=1}^{N_{t}} ∫_{τ_{n}−ε}^{τ_{n}+ε} f(X_{s}^{a,ε})|Ṡ_{s}^{a,ε}|ds.  (37)

Observe that for τ_{n} − ε < s < τ_{n} + ε one has

Ṡ_{s}^{a,ε} = \frac{1}{ε}ψ(\frac{s − τ_{n}}{ε} )ΔX_{τ_{n}}

so that

∫_{τ_{n}−ε}^{τ_{n}+ε} f(X_{s}^{a,ε})|Ṡ_{s}^{a,ε}|ds = \frac{1}{ε} ∫_{τ_{n}−ε}^{τ_{n}+ε} f(X_{s}^{a,ε})ψ(\frac{s − τ_{n}}{ε} )|ΔX_{τ_{n}}| ds.

Making the change of variables z = (s − τ_{n})/ε in each integral, we obtain

∫_{0}^{t} f(X_{s}^{a,ε})|Ṡ_{s}^{a,ε}|ds = ∑_{n=1}^{N_{t}} |ΔX_{τ_{n}}| ∫_{−1}^{1} f(X_{τ_{n}+εz}^{a,ε})ψ(z)dz.

To compute the limit as ε → 0 in the right hand member of the last equality, use that

X_{τ_{n}+εz}^{a,ε} → X_{τ_{n}}^{a} ∫_{−1}^{z} ψ(w)dw + X_{τ_{n}−}^{a} ∫_{z}^{1} ψ(w)dw.

This proves the statement.

References

[1] Azaïs, J-M. and Wschebor, M. Almost sure oscillation of certain random processes, Bernoulli 2 (3), 1996, 257–270.

[2] Azaïs, J-M. and Wschebor, M. Oscillation presque sûre de martingales continues, Séminaire de Probabilités XXXI, Lecture Notes in Mathematics 1655, Springer, 1997, 69–76.

[3] Bertoin, J. Lévy Processes. Cambridge University Press, Cambridge, 1966.

[4] Berzin, C. and León, J.R. Weak convergence of the integrated number of level crossings to the local time of the Wiener process. Comptes R. Acad. Sc. Paris, Sér. I 319, 1994, 1311–1316.

[5] Billingsley, P. Convergence of probability measures. Wiley, New York, London, Sydney, 1968.

[6] Borodin, A.N. and Ibragimov, I.A. Limit theorems for functionals of random walks. (Predel’nye teoremy dlya funktsionalov ot sluchajnykh bluzhdanij.) (Russian) Trudy Matematicheskogo Instituta Imeni V. A. Steklova.

195. Nauka, Sankt-Peterburg, 1994. (See also: Borodin, A.N. and Ibragimov, I.A. Limit theorems for functionals of random walks. (English) Proceedings of the Steklov Institute of Mathematics. 195. American Mathematical Society (AMS), Providence, RI, 1995.)

[7] Dacunha-Castelle, D. and Florens-Zmirou, D. Estimation of the coefficient of a diffusion from discrete observations, Stochastics 19, 1986, 263–284.

[8] Florens-Zmirou, D. On estimating the difussion coefficient from discrete observations, J. Appl. Prob. 30, 1993, 790–804.

[9] Génon-Catalot, V. and Jacod, J. On the estimation of the diffusion coefficient for multidimensional diffusion processes, Ann.Inst. H.Poincaré, Probab. Stat. 29, 1993, 119–151.

[10] Jacod, J. Rates of convergence to the local time of a diffusion. Ann.Inst. H.Poincaré, Probab. Stat. 34, 1998, 505–544.

[11] Jacod, J. Non-parametric kernel estimation of the diffusion coefficient of a diffusion. Scand. J. Statist. 27 (1), 2000, 83–96.

[12] Jacod, J. and Shiryaev, A.N. Limit Theorems for Stochastic Processes. Springer, Berlin, Heidelberg, 1987.

[13] Kurtz, T. G. and Protter, P. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 (3), 1991, 1035–1070.

[14] Perera, G. and Wschebor, M. Crossings and occupation measures for a class of semimartingales, Ann. Probab. 26 (1), 1998, 253–266.

[15] Perera, G. and Wschebor, M. Inference on the Variance and Smoothing of the Paths of Diffusions, Ann.Inst. H.Poincaré, Probab. Stat. 38 (6), 2002, 1009–1022.

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

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