A STROOCK FORMULA FOR A CERTAIN CLASS OF LÉVY PROCESSES AND APPLICATIONS TO FINANCE

We find a Stroock formula in the setting of generalized chaos expansion introduced by Nualart and Schoutens for a certain class of Lévy processes, using a Malliavin-type derivative based on the chaotic approach. As applications, we get the chaotic decomposition of the local time of a simple Lévy process as well as the chaotic expansion of the price of a financial asset and of the price of a European call option. We also study the behavior of the tracking error in the discrete delta neutral hedging under both the equivalent martingale measure and the historical probability.


Introduction
In general, a Lévy process has not the chaotic representation property (say CRP for brevity) but Nualart and Schoutens have developed in [8] a kind of generalized CRP for a large class of Lévy processes. This work enabled to define a Malliavin-type derivative using the chaotic approach in a recent paper of Léon et al. [6]. The main goal of the present article is to get a Stroock formula in this setting. This formula gives the kernels of the chaotic decomposition of smooth random variables as functionals of the underlying Lévy process. For a complete survey on Lévy processes, we refer to [2,14].
We apply this formula to obtain the chaos decomposition of some functionals of simple Lévy processes. These processes are the sum of a Wiener process and m independent Poisson processes. As it is pointed by the name, they are easy to handle and very useful for doing simulations since they approximate the square integrable compound Poisson process in the L 2 (Ω × [0,T]) sense (see [6]). In particular, they approximate the process given by the sum of jumps with size greater than ε > 0.
In this paper, firstly, we obtain the decomposition of the local time L(t,x) defined as the density of the occupation measure. Secondly, taking account that these simple Lévy processes have been studied in [6] for pricing and hedging options in financial markets driven by such processes, we apply the Stroock formula to obtain the chaos expansion of an asset price as well as the price of a European call option based on this asset. The chaotic approach enable us to study the asymptotic behavior of the variance, since the terms of the chaotic expansion are orthogonal and in particular uncorrelated, and this is useful in practical hedging.
The paper is organized as follows. The second section is devoted to recall some definitions and results related to Lévy chaotic calculus as well as to give some new remarks. In the third section, we get the Stroock formula. The fourth section is devoted to apply it in order to obtain the chaotic expansion of the local time of a simple Lévy process. Finally, the last three sections discuss chaotic expansions for the price of a financial asset and the price of a European call option, in a simple jump-diffusion model as well as the asymptotic behavior of the variance of the tracking error for discrete delta neutral hedging with respect to the mesh of the subdivision and under both the equivalent martingale measure and the historical probability.

Basic elements of Lévy chaotic calculus
2.1. The Teugels martingales family associated to a Lévy process. Let X = {X t : t ≥ 0} be a real-valued Lévy process defined on a complete probability space (Ω,Ᏺ,P). Henceforth, we always assume that we are using the càdlàg version. Let {Ᏺ t : t ≥ 0} be the natural filtration of X completed with the P-null sets of Ᏺ. We also assume that the Lévy measure ν of X satisfies the following condition: there exist ε > 0 and δ > 0 such that (−ε,ε) c e δ|x| ν(dx) < +∞, (2.1) where (−ε,ε) c stands for the complement of the interval (−ε,ε). This implies that X t has moments of all orders and that the polynomials are dense in L 2 (R,P • X −1 1 ) (see [8]). Define We have the following. (i) The processes X (i) = {X (i) t : t ≥ 0}, i = 1,2,..., are Lévy processes that jump at the same points as X.
are martingales. The predictable quadratic covariation process of Y (i) and Y ( j) is given by Now, we introduce the so-called Teugels's martingales, Moreover, we have the following.
(iv) The processes {H (i) t : t ≥ 0} are martingales with predictable quadratic variation process given by where

Iterated integrals and a generalized CRP. Let
We remark that all these integrals are well defined since all the processes H (i) for i ≥ 1 are martingales with respect to the filtration {Ᏺ t : t ≥ 0}. Remark also that these iterated integrals are not the usual multiple stochastic integrals.
As a consequence of the strong orthogonality of the family of Teugels martingales, we have the following proposition.
Proposition 2.2. Let F ∈ L 2 (Ω,Ᏺ,P). Then F has a unique representation of the form where f i1,...,in ∈ L 2 (Σ n ). (2.13) and i means that the ith index is omitted.

Derivative operators
we define the spaces of the random variables that are differentiable in the th direction. For this, we define the following subset of L 2 (Ω): (2.14) Observe that, as in the classical situation for Gaussian processes, D ( )1,2 is dense in L 2 (Ω), since the elements of L 2 (Ω) with a finite chaotic expansion are in D ( )1,2 .
Observe that because the independence of two Poisson processes adapted to the same filtration implies that they do not jump at the same time (see [2]). Since .., N m t since the uniqueness of the CRP in terms of the W t , N 1 t ,..., N m t . To unify the notations, we will write G 0 (t) = W t and G j (t) = N j t for j = 1,...,m. Also, we will denote by L (i1,...,in) n ( f ) the iterated integral of f with respect to G i1 ,...,G in : (2.21) Thus, we have also the chaotic representation property in terms of the G i 's. where f i1,...,in ∈ L 2 (Σ n ).
In this setting, we recall a result of Léon et al. [6] which says that is possible to compute the derivatives in the directions W,N 1 ,...,N m ,..., following the classical rules on each space. Recall that in Poisson setting the derivative is a difference operator (see [10]).
We use the space where Ω 0 is the canonical space of the Wiener process and Ω i , i = 1,...,m, are, respectively, the canonical spaces of Poisson processes N i , i = 1,...,m, that are the spaces of all possible paths of Poisson's. Remark 2.6. If we iterate the derivative with respect to the Poisson's i and j (i < j), we obtain but this equality is not true in general if we only interchange the superindexes or the subindexes. If the iteration is done with respect to the Then k s = #I s will be the order of derivation with respect to G s , and hence m i=0 k i = n and ∪ m i=0 J i = {t 1 ,...,t n }. Therefore, with the convention that D i,0 Ji is the identity, and we get (2.25) Remark 2.9. If there's only one Poisson process and no Brownian part, different authors as Léon et al. [7] and Privault [12] have used the iterated derivatives to find the chaotic decomposition of functionals of the d first jump times of the process. In the present article, we consider different Poisson processes and a Brownian part. Moreover, we are not restricted to a specified finite number of jump times.

Stroock formula for Lévy processes
The aim of this section is to derive a Stroock formula for functionals of a Lévy process belonging to D ∞ . First we deal with the case that F is an element of a specific chaos and finally we will extend the result to F in D ∞ . We start with the following lemma.
Proof. First remark that, in the sum, only one term is different from zero. To prove the lemma, we use induction on n.
For n = 2, we have and applying now the operator D j2 t2 , we get we get the following equality: Now, we apply Lemma 3.1 to obtain the Stroock formula.
Proof. As the iterated derivatives of order n of elements belonging to chaos of order less than n are zero and applying the above lemma, we get where M n is a sum of variables in chaos of order greater than or equal to n. Taking the expectation, we obtain the desired formula.

Chaos expansion of the local time of a simple Lévy process
The aim of this section is to apply the Stroock formula to find the chaotic decomposition of the local time of a simple Lévy process. We denote by H n the nth Hermite polynomial defined by for n ≥ 1 and H 0 (x) = 1.

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We define the local time L(t,x) of a Lévy process X as the density of the occupation measure It is well known that this density L(t,x) exists and is a nondecreasing function of t and the measure L(dt,x) is concentrated on the level set {s : X s = x}. Moreover, Barlow [1] shows that L(t,x) has an almost surely jointly continuous version.
It is well known also that we can write To apply the Stroock formula, we consider where p ε is the centered Gaussian kernel with variance ε > 0. The classical idea of approximating the Dirac distribution δ x by p ε has been used to calculate the chaotic decomposition of the local time in the case of the Brownian motion by Nualart and Vives [11] and for the fractional Brownian motion by Coutin et al. [3] and Eddahbi et al. [4]. Before stating precise results of this section, we prove some technical lemmas.
Proof. We only apply Remark 2.8 of Section 2.
Lemma 4.2. Set g n (y) = H n (y)exp(−y 2 /2). Then Interchanging the derivative operator and the expectation, and using we get the formula of the lemma.
Then, we can prove the following proposition.
We apply the Stroock formula to p ε (σW s + m i=1 α i N i s − x) and use Lemma 4.1 for the expression of the iterated derivative and Lemma 4.2 for the expectation with respect to the Wiener part. Besides, we calculate the expectation with respect to the Poisson parts.
Remark 4.5. Note that the kernels depend only on k 0 ,k 1 ,...,k m . Hence, it is the same for all sets of indexes i 1 ,...,i n that derive to equal k 0 ,k 1 ,...,k m . Observe also that although the kernel is the same, the iterated integral L i1,...,in n can depend on the order of the indexes as we will see in the next example: where T i ,i ∈ N, are the jump times of Poisson. But In order to establish the chaotic expansion of the local time of a simple Lévy process we state the following lemma (see [11]). Lemma 4.6. Let {F ε } ε>0 be a family of square integrable random variables with the expansions where f ε i1,...,in belongs to L 2 ([0,∞) n ).

Assume that
(i) f ε i1,...,in converges in L 2 ([0,∞) n ), when ε goes to zero, to some symmetric function Then we can prove the following proposition.  Proof. We must check the two hypotheses of Lemma 4.6. We start with (ii    Hence, the general term of the right-hand side of (4.22) behaves as ck −3/2 0 and the corresponding series is convergent.
Note that in our setting q 0 = ··· = q m = 1, because we do not work with the H (i) 's but use directly the Wiener and Poisson processes. Now, it remains to check Lemma 4.6 (i). We have

(4.25)
It is clear that f ε i1,...,in converges to f i1,...,in pointwise and using the dominated convergence theorem, we see easily that condition (i) holds.
Finally, we will show, following standard arguments, that the limit of L(t,x). The above estimates are uniform in x ∈ R. Therefore, we can conclude that the convergence of x)ds to Λ x t holds in L 2 (Ω × R, P ⊗ µ), for any finite measure µ. As a consequence, for any continuous function g in R with compact support, we have that Hence, (4.28) which implies that Λ x t = L(t,x).

Chaos expansion of the price of a financial asset
In this section, we will use the Stroock formula to get the chaotic decomposition of the asset price S t driven by the Lévy process X t = µt + σW t + m j=1 α j N j t , where N j t , j = 1,...,m, are independent compensated Poisson processes with parameters λ j , j = 1,...,m. This means that S t satisfies the stochastic differential equation where c = µ − σ 2 /2 − m i=1 α i λ i and β i = log(1 + α i ) for i = 1,...,m with the condition α i > −1 to guarantee the positivity of the price (see [13]).
Remark 5.1. If τ is a jump time of the Poisson process N j , the relative jump (S τ − S τ− )/S τ is equal to α j and the absolute jump log(S τ ) − log(S τ− ) of the log price process log(S t ) is β j . Now we compute the iterated derivatives of S t and we obtain  j , 0< ξ < t, (5.11) which tends to 0 if p tends to infinity or t tends to zero. In the case of the Brownian motion, γ j = α j = 0 for j = 1,...,m and t n σ 2n n! = s 2 0 e 2tµ e tσ 2 − 1 . (5.12)

Chaos expansion of the price of a European call option
Let U t be the price on t of a European call option written on the asset described in the last section. By the no-arbitrage theory, where Q is the unique martingale measure described in [6] and g(x) = x ∨ 0. Note that in order to get the uniqueness of Q, we have to add m additional assets defined by equations with the condition that there exist constants M,L 1 ,...,L m , with L j ≤ λ j , such that where r > 0 is the fixed interest rate and B is a matrix given by Using Girsanov's theorem, we find a Q-Brownian motion W Q and m independent Q-Poisson processes with parameters λ j = λ j − L j , N Q,1 ,...,N Q,m , such that we can write The explicit solution of (6.5) is where we have used the known commutative property between the derivative operator and the conditional expectation (see [10]). Now, we have to compute E Q [D i1,...,in s1,...,sn g(S T − K)]. In order to do this, we approximate the function g, uniformly, by the sequence of Ꮿ ∞ functions: (6.9) where Φ ε is the cumulative probability function of the centered normal law with variance ε.
Observe that the first derivative of g ε is Φ ε and the high-order derivatives are the functions p ( j−2) ε (x) for j ≥ 2, that are for every j, the (j − 2)-order derivatives of the centered Gaussian kernel of variance ε, denoted by p ε .

A Stroock formula for Lévy processes and applications
Let F ε = g ε (S T − K). Clearly, F ε converges in L 2 to F = (S T − K) + = U T . Then, applying Remark 2.8 to g ε and assuming s 1 < s 2 < ··· < s n < T, where S T = S T e m j=1 βj j T . Using the formula f (ae bx − c) (n) = n j=1 f ( j) (ae bx − c)b n a j e jbx c n, j with c n,n = c n,1 = 1, we get the following.
(i) If k 0 = 0, Therefore, we have to compute and pass to limit, when ε tends to zero, three types of expectations: In the first case, using the uniform convergence of g ε to g, we obtain as a limit For the second case, using the dominated convergence theorem, we have that the limit is In order to get the expectations with respect to the Gaussian part, we need only the fact that and the following lemma.
We have Finally, for the expectations of third type, we need the following result.
with f being the density of the log-normal law with parameters log(C) and σ √ T.
Proof. We have Taking h (x) = x +2 f (x) and integrating by parts, using that h and all its derivatives are fast decreasing functions, the last integral is equal to where * denotes the convolution. Using that p ε is an approximation of the identity, we obtain the limit  where now C reduces to s 0 exp{(r − (σ 2 /2))T}.

Chaos expansion of the tracking error in discrete delta-neutral hedging
Assume that we are in a neutral risk environment. That is, from now on, we are concerned with the unique probability Q that makes e −rt S t a martingale. Here we follow some ideas from [5]. Let U t be the prize of a derivative security. Its actualized price U t is given by By the Markov property of S t , the process U t can be written as a function of t and S t . Moreover thanks to (6.6), we can write where p i (y) = a i,i x i + a i,i−1 x i−1 + ··· + a i,1 x. Note that there are only m + 1 processes H i associated to Y which are different from zero (see [6]). In our case the explicit expression of the right-hand side of the last equation is a martingale plus the following term: where τ β u(t,x) = u(t,x + β) − u(t,x). Hence, the martingale property of u(t,Y t ) is equivalent to the condition that u satisfies the following partial differential integral equation: where u 0 is a given function.  Of course, this is a centered random variable that converges to zero when the partition becomes finer and finer but, moreover, it has no first chaos. In fact