Malliavin derivative of random functions and applications to L\'evy driven BSDEs

We consider measurable $F: \Omega \times \mathbb{R}^d \to \mathbb{R}$ where $F(\cdot, x)$ belongs for any $x$ to the Malliavin Sobolev space $\mathbb{D}_{1,2}$ (with respect to a L\'evy process) and provide sufficient conditions on $F$ and $G_1,\ldots,G_d \in \mathbb{D}_{1,2}$ such that $F(\cdot, G_1,\ldots,G_d) \in \mathbb{D}_{1,2}.$ The above result is applied to show Malliavin differentiability of solutions to BSDEs (backward stochastic differential equations) driven by L\'evy noise where the generator is given by a progressively measurable function $f(\omega,t,y,z).$


Introduction
Backward stochastic differential equations (BSDEs) have been studied with growing interest and from various perspectives. They appear in stochastic control theory, as Feynman-Kac representation of second order semilinear PDEs, and have many applications in Finance and Insurance (see, for instance, El Karoui et al. [16], the survey paper from Bouchard et al. [11] or Delong [13], and the references therein). Pardoux and Peng have considered in [22] and [23] BSDEs of the form where W denotes the Brownian motion. Under suitable smoothness and boundedness conditions on the coefficients they have shown that the two-parameter process D θ Y s is a.s. continuous in s ∈ [θ, T ] and, moreover, {D θ Y θ := lim s↓θ D θ Y s : θ ∈ [t, T ]} is a version of the process {Z s : s ∈ [t, T ]}. In this way, using the relation it is possible to represent Z (with the right interpretation) as These representations turned out to be useful in regularity estimates for Y and Z which play an important role for estimates of convergence rates of time-discretizations (see, for example, [10], [12], [11], [14]). El Karoui et al. [16] generalized this result to progressively measurable generators (ω, t) → f (ω, t, y, z). Delong and Imkeller considered Malliavin differentiability of delayed BSDEs driven by Lévy noise in [14]. In this paper, we first consider a measurable function F : Ω × R d → R where F (·, x) belongs to the Malliavin Sobolev space D 1,2 for any x ∈ R d . We ask for sufficient conditions on F and G 1 , . . . , G d ∈ D 1,2 such that F (·, G 1 , . . . , G d ) ∈ D 1,2 . Our aim was to find very general conditions such that the result is also applicable for BSDEs with non-Lipschitz generators. As we work in the Lévy setting, the results hold of course especially for the Brownian case. In this respect, we could generalize somewhat the conditions on the generator given in [16].
The paper is organized as follows: Section 2 contains the setting and a collection of used notation. Section 3 starts with the definition of the Malliavin derivative in the Lévy space. We recall a method introduced in [25] which allows to compute the Malliavin derivative D t,x for x = 0 without knowing the chaos expansion and without imposing the condition that the underlying probability space is the canonical Lévy space [28]. Based on the fact that D t,x for x = 0 and D t,0 are of different nature we solve the question about the Malliavin differentiability of F (·, G 1 , . . . , G d ) ∈ D 1,2 in two steps: In Subsection 3.3 we treat the question concerning D t,0 . Here we use the result from [29] that for the Brownian motion the Malliavin Sobolev spaces D W 1,p (E) (E denotes a separable Hilbert space and p > 1) coincide with the Kusuoka-Stroock Sobolev spaces which are defined using the concept of ray absolute continuity and stochastic Gateaux differentiability. In Section 4 we formulate the conditions on the BSDE such that it is Malliavin differentable, present the proof and give an example.

Setting
Let X = (X t ) t∈[0,T ] be a Lévy process on a complete probability space (Ω, F , P) with Lévy measure ν. We will denote the augmented natural filtration of X by (F t ) t∈[0,T ] and assume that F = F T . The Lévy-Itô decompositon of a Lévy process X can be written as where σ ≥ 0, W is a Brownian motion andÑ is the compensated Poisson random measure corresponding to X. Throughout the paper we will use the notation X(ω) = (X t (ω)) t∈[0,T ] , which is a càdlàg trajectory. Let ∆X given by ∆X t := X t − lim sրt X s denote the process of the jumps of X.
We close this section with a list of notation for càdlàg processes on the path space and for BSDEs. • For a measurable mapping Y : • For a fixed t ∈ [0, T ] the notation induces the natural identification By this identification we define a filtration on this space by where N X [0, T ] denotes the null sets of B (D[0, T ]) with respect to the image measure P X of the Lévy process X. For more details on D[0, T ], see [8] and [15,Section 4].
• | · | denotes a norm in R n .
• For later use we recall the notion of the predictable projection of a stochastic process with parameters.
According to [26,Proposition 3] (see also [20,Proposition 3] or [2, Lemma 2.2]) for any z ∈ L 2 (P⊗Ñ) : such that for any fixed x ∈ R the function ( p z) ·,x is a version of the predictable projection (in the classical sense) of z ·,x . In the following we will always use this result to get predictable projections which are measurable w.r.t. a parameter. Again, we call p z the predictable projection of z.

Malliavin calculus
3.1 Definition of D 1,2 using chaos expansions The random measure M defined in (2) allows to introduce the Malliavin derivative defined via chaos expansions (see, for example, [27]) as follows: Any ξ ∈ L 2 has a unique chaos expansion (see [17,Theorem 2]) and it holds , and I n denotes the n-th multiple integral with respect to M from (2). The multiple integrals with respect to M can be defined as follows: If n = 0 set L 0 2 := R and I 0 (f 0 ) := f 0 for f 0 ∈ R. For n ≥ 1 we start with a simple function f n ∈ L n 2 given by where the sets B k i ∈ B([0, T ] × R) for k = 1, . . . , m, i = 1, . . . , n are disjoint for fixed k, and Ñ(B k i ) < ∞ for all i and k. Then By denseness of these simple functions in L n 2 and by linearity and continuity of I n , one extends the domain of the n-fold multiple stochastic integral I n to become a mapping I n : L n 2 → L 2 . It holds I n (f n ) = I n (f n ) wheref n denotes the symmetrization of f n w.r.t. the n pairs of variables in [0, T ] × R. For f n ∈ L n 2 and g m ∈ L m 2 we have The space D 1,2 consists of all random variables ξ ∈ L 2 such that The Malliavin derivative is defined for ξ ∈ D 1,2 by We will also consider and Then, for σ > 0 and ν = 0 :

From canonical to general probability spaces
Sole et al. introduced in [28] the canonical Lévy space and proved that for x = 0 the Malliavin derivative D r,x ξ (defined via chaos expansions) equals in this space an increment quotient. We will discuss here how to transfer results from the canonical Lévy space to general probability spaces carrying a Lévy process.
Assume (Ω 1 , F 1 , P 1 ) , (Ω 2 , F 2 , P 2 ) to be probability spaces with Lévy processes such that X i corresponds to a given Lévy triplet (γ, σ, ν) for i = 1, 2. Furthermore, assume that F i is the completion of the σalgebra generated by X i . For the processes X 1 , X 2 , we get the associated independent random measures M 1 and M 2 like in (2), and the families of multiple stochastic integrals I 1 n (f n ) n∈N , I 2 n (f n ) n∈N , respectively. The following assertion is taken from [25,Corollary 4.2], where it is formulated for Lévy processes with paths in D[0, ∞[.
and suppose that these random fields have chaos decompositions for f n , g n being functions in L 2 (E, E, ρ)⊗L n 2 which are symmetric in the last n variables, where '⊗' denotes the Hilbert space tensor product. Assume that for ρ-almost all e ∈ E there are functionals Then for all n ∈ N it holds f n = g n , ρ ⊗ Ñ ⊗n -a.e.
Roughly speaking, if we have the same functional acting on Lévy processes X i defined on the probability spaces (Ω i , F i , P i ) for i = 1, 2 then the deterministic kernels of their chaos expansions coincide.
The Factorization lemma (see, for instance, [5, Section II.11]) implies that for any ξ ∈ L 2 there exists a measurable functional g ξ : D([0, T ]) → R such that for a.a. ω ∈ Ω. Sole et al. consider in [28] the Malliavin derivative which corresponds to the chaos expansion with respect to the independent random measure σdW t δ 0 (dx) + xÑ (dt, dx). In this paper we use the equivalent approach without multiplying the Poisson random measure with x. Hence the according Malliavin derivative for x = 0 and M from (2) is just a difference instead of the difference quotient from [28]. Applied on g ξ (X(ω)) this gives in the canonical space for P ⊗ Ñ a.e. (ω, r, x). The following characterization that g ξ (X) ∈ D R 0 and it holds then for x = 0 P ⊗ Ñ-a.e.
In the situation of the previous lemma, one may ask whether properties of g ξ (X) that hold P-a.s. are preserved P ⊗ Ñ-a.e. for g ξ (X + x1I [t,T ] ). The positive answer is given by the following result (the proof can be found in the appendix).  (ii) Let ξ = g ξ (X) ∈ L ∞ (Ω). By the same reasoning as in (i), it follows from Note that the boundedness of g ξ (X + v1I [r,T ] ) implies boundedness of the difference (7), which -in case of L 2 integrability -equals the Malliavin derivative for v = 0.

Malliavin calculus for random functions
We want to address the following problem. Let be jointly measurable, for any y ∈ R d we assume F (·, y) ∈ D 1,2 , and for a.a.
We will deal with the problem in two steps: First we will find conditions on F and G such that separately and then use relation (5).

The case F
For the sufficiency we refer to [25,Theorem 5.2]. There the proof is carried out only for d = 1 but it is easy to see that the multidimensional case can be proved in the same way.
For the necessity we consider F, G 1 , . . . , G d as given by functionals g F (·, y), g G 1 , . . . , g G d and conclude from Lemma 3.2 that

Hence expression (8) equals in fact
where we have used Lemma 3.2 again.

The case
As it is known from the Lévy-Itô decomposition that the Brownian part and the pure jump part of a Lévy process are independent we may represent X on the completion of ( is the space of continuous functions starting in 0, and F W is the completed Borel σ-algebra with respect to the Wiener measure P W . To work on the canonical space (Ω W , F W , P W ) we continue with a short reminder on Gaussian Hilbert spaces and refer the reader for more information to Janson [18]. Consider the Gaussian Hilbert space H :

Because of Itô's isometry we may identify H with
The space The main idea to get sufficient conditions for F (·, G 1 , ..., G d ) ∈ D 0 1,2 consists in applying Theorem 3.10 below. We proceed with a collection of definitions and some facts related to this theorem. Definition 3.6 ([29], [21]). Let 1 ≤ p < ∞ and S ⊆ D 1,p (P W ) be a dense set of smooth random variables and E a separable Hilbert space. By D W 1,p (E) we denote the completion of [1]). This means we may reformulate the question of this section by asking for sufficient conditions such that . The answer will be Theorem 3.12 at the end of this section.
Let E 1 and E 2 be separable Hilbert spaces. A bounded linear operator A : [9]). We will denote by HS(H 0 , E) the space of Hilbert-Schmidt operators between H 0 and E. Definition 3.7 ([18], [9]). With L 0 (P W ; E) we denote the space of E-valued random variables, equipped with the topology of convergence in probability. For ξ ∈ L 0 (P W ; E) and h ∈ H 0 we define the Cameron-Martin shift by One of the properties of the Cameron-Martin shift is the Cameron-Martin formula.
(For an integral of E-valued objects, we always use the Bochner integral.) h for h ∈ H 0 , and the Radon-Nikodym derivative is given by (ii) Analogously to the proof of Theorem 14.1 (vi) in Janson [18] for 1 ≤ q < p we choose r = p p−q so that 1 r + q p = 1, and by the Cameron-Martin formula and Hölders inequality we get (iii) This is Theorem 14.1 (viii) of [18] for E-valued random variables (in Remark 14.6. of [18] it is stated that Theorem 14.1 which is formulated for real valued random variables holds also for random variables with values in a separable Banach space).

Definition 3.9 ([18]
, [9]). (i) A random variable ξ ∈ L 0 (P W ; E) is absolutely continuous along h ∈ H 0 (h-a.c.) if there exists a random variable ξ h ∈ L 0 (P W ; E) such that ξ h = ξ a.s. and for all ω W ∈ Ω W the map is absolutely continuous on bounded intervals of R.
We will also need the following result.
Proof. For E = R this is Theorem 15.21 of [18]. One can generalize the proof to E-valued random variables since by the Radon-Nikodym property of E (see [4]), the fundamental theorem of calculus holds for absolutely continuous functions if Bochner integrals are used.
Step 1. We will use the characterization of D W 1,p (E) from Theorem 3.10. In fact, we will prove for any u ∈ R and h ∈ H 0 the relations where the first equation is E-valued with G = (G 1 , . . . , G d ), and the second equation is scalar with ∇ y = ( ∂ ∂y 1 , . . . , ∂ ∂y d ).
Since by assumption (iv) we infer that |u| −|u| ρ sh (∂ h F )(·, G) E ds < ∞, P-a.s and according to Theorem 3.11 it follows from the first line of (9) that F (·, G) is r.a.c. From the second line of (9) we get that F (·, G) is s.G.d. and ). Together with Theorem 3.10 this would imply the assertion of the theorem. So it remains to show the relations in (9) which will be done in Steps 2 and 3.
Hence by Theorem 3.11 for each u ∈ R and h ∈ H 0 the E-valued equation holds for all ω W up to an exception set C y ∈ F W with P W (C y ) = 0. Consequently, for each u ∈ R and h ∈ H 0 we have the real-valued equation (where we use the notation ρ uh (ω) = (ω W + ug h , ω J )) for all ω with the exception of a setC y ∈ F with P(C y ) = 0. Since the LHS is a.s. continuous in y, we can find an exception setC ∈ F , P(C) = 0, independent of y provided we can show that also the RHS is continuous in y. To do this we estimate for y,ỹ ∈ B N (0) Since by Lemma 3.8 u 0 ρ sh K N ds < ∞, P-a.s., it follows that for a.a. ω the RHS of (10) is continuous in y. Consequently, on Ω\C ∈ F relation (10) is true for all y ∈ R d . Putting the terms to zero onC, the right hand side of (10) is jointly measurable w.r. t. (ω, y). We may replace y by G(ω) := (G 1 (ω), . . . , G d (ω)) and get Step 3 We show that F (·, G) is r.a.c. For this we choose an interval [0, t 1 ], t ∈ [0, t 1 ], let 0 = s 0 < s 1 < ... < s n = t 1 and consider for s t k := s k ∧ t the expression For any b := s t k and a := s t k−1 we derive from (11) and the mean-value theorem that a.s.
for some θ ∈ [0, 1]. We may write the second term because F (ω, y) is C 1 w.r.t. y. Similarly to (10) , for each G l ∈ D W 1,q (E), we have for all t ∈ R and h ∈ H 0 that To obtain (9) we rewrite (12) in the following way remainder terms.
The remainder terms are given by where we use ∂ h G(ω) := (D W G)(ω), h H 0 as an abbreviation. It is sufficient to show that the sum of the remainder terms tends in probability to zero for (s t k ) := max 1≤k≤n |s t k − s t k−1 | → 0. Because of (13), for arbitrary ε 1 > 0 one can choose (s t k ) sufficiently small such that

Since Lemma 3.8 (ii) implies that
The term I 2 we estimate similarly: For arbitrary ε 2 > 0 we infer from assumption (i) that for sufficiently small (s t k ) For the remaining integral I 3 we proceed as follows: By Lemma 3.8 (iii) the map is uniformly continuous, which then also holds for

Malliavin derivative of solutions to BSDEs
In this section we apply our results on Malliavin differentiability of random functions to generators of BSDEs. We state in Theorem 4.4 that under conditions on the smoothness of the data (ξ, f ) solutions to BSDEs are Malliavin differentiable. For simplicity, we set σ = 1 in (1). The results hold true (with the appropriate modifications) if at least one of them, σ or the Lévy measure ν, are non-zero.
For 0 ≤ t ≤ T we consider the BSDE for measurable u : R 0 → R (the conditions on g and g 1 formulated in (A f g) below ensure that the expression is well-defined). For shortness of notation, we define For the terminal value ξ and the function f g we agree upon the following assumptions: s. and f satisfies the following Lipschitz condition: There exists a constant L f such that for all t ∈ [0, T ], η,η ∈ R 3 |f (X, t, η) − f (X, t,η) | ≤ L f |η −η| 1 .

Assumption (A f f) is, in fact, stronger than needed in Theorem 4.4 below.
It is enough to require that |(D s,x f ) (t, G)| ≤ Γ s,x , P ⊗ Ñ-a.e. holds for the solution G = (Y t , Z t , U t ) and for the members G = (Y n t , Z n t , U n t ) of the approximating sequence appearing in the proof of Theorem 4.4.

3.
The assumption (A f g) on g can be extended to a dependency on t and ω.
Also g 1 may be assumed to be time-dependent. To keep the same proof of Theorem 4.4 feasible, we have to impose conditions (A f a-f) on g (with R 3 replaced by R as g is then a random process with one parameter). Furthermore, we have to assume that g 1 : To cover the issue of existence of solutions to BSDEs we refer to the following result:  . Assume that (ξ, f g ) and (ξ ′ , f ′ g ) satisfy ξ, ξ ′ ∈ L 2 and suppose the generators fulfill (A f a-c), while g is Lipschitz and g 1 ∈ L 2 (R 0 , B(R 0 ), ν). Then there exists a constant C > 0 such that for the corresponding solutions (Y, Z, U) and (Y ′ , Z ′ , U ′ ) to (14) it holds We state now the result about the Malliavin derivative of solutions to BSDEs. For the proof we apply Itô's formula like in the original work due to Pardoux and Peng [22]. The benefit is that one does not need any higher moment conditions on the data than L 2 . Hence this result is a generalization of El Karoui et al. [16]. where (ii) For the solution (Y, Z, U) of (14) it holds and D r,y Y admits a càdlàg version for Ña.e. (r, y) We present an example of an FBSDE where we specify the dependence on ω in the generator by a forward process such that (A f f) holds.

Example 4.5.
Consider the case of a Lévy process X such that E|X t | 2 < ∞ for all t ∈ [0, T ]. Assume the generator to be of the type f (s, ω, y, z, u) =f (s, Ψ s (ω), y, z, u), withf having a continuous partial derivative in the second variable bounded by K. Let Ψ denote a forward process given by the SDE Then condition (A f f) is satisfied under the requirements (i) The functions b : R → R and σ : R → R are continuously differentiable with bounded derivative.
and is continuously differentiable in ψ with bounded derivative for fixed This follows, since (D s,x f ) (t, G) is given by Thus we may choose Γ = C sup s∈[0,T ] |DΨ s |, where C depends on K, C β and the Lipschitz constants for b, σ and β.

Proof of Theorem 4.4
Let us start with a lemma providing estimates for the Malliavin derivative of the generator.
Moreover, for G ∈ (D 1,2 ) 3 it holds f (X, t, G) ∈ D 1,2 and Proof. According to Corollary 3.4 we may replace X by X + v1I [r,T ] and get from the Lipschitz property (A f c) that From (A f f ) one concludes then (19).
For v = 0 we apply Lemma 3.2 to get and hence (20) follows from (19). In the case of v = 0, by assumption (A f e) we may apply Theorem 3.12. Thus we get the Malliavin derivative (20) follows from conditions (A f c) and (f) using that the partial derivatives are bounded by L f .

Proof of Theorem 4.4.
The core of the proof is to conclude assertion (ii) which will be done by an iteration argument. To simplify the notation we do not mention the dependency of f on X in most places.
(i) For those (r, v) such that D r,v ξ ∈ L 2 the existence and uniqueness of a solution (Y r,v , Z r,v , U r,v ) to (16) follows from Theorem 4.2 since F r,v meets the assumptions of the theorem.
(ii) By Theorem 4.3 the solution depends continuously on the terminal condition and Dξ is measurable w.r. t. (r, v). We infer the measurable dependency (r, v) → (Y r,v , Z r,v , U r,v ) as follows: Since by Theorem 4.3 the mapping is continuous one can show the existence of a jointly measurable version of by approximating Dξ with simple functions in L 2 (P ⊗ Ñ). Joint measurability (for example for Z) in all arguments can be gained by identifying the spaces The quadratic integrability with respect to (r, v) also follows from Theorem 4.3 since ξ ∈ D 1,2 .
Using an iteration scheme, starting with (Y 0 , Z 0 , U 0 ) = (0, 0, 0), we get Y n+1 by taking the optional projection which implies that The process Z n+1 given by one gets by the martingale representation theorem w.r.t. M (see, for example, [3]): Step 1. It is well-known that (Y n , Z n , U n ) converges to the solution (Y, Z, U) in L 2 (W ) × L 2 (W ) × L 2 (Ñ ). Our aim in this step is to show that Y n , Z n and U n are uniformly bounded in n as elements of L 2 (λ; D 1,2 ) and L 2 (λ ⊗ ν; D 1,2 ), respectively. This will follow from (27) below.
Since by [3,Theorem 4.2.12] the process ]0,t]×R D r,v Z n+1 s,x M(ds, dx) t∈[0,T ] admits a càdlàg version, we may take a càdlàg version of both sides. By Itô's formula (see, for instance, [3]), we conclude that for 0 < r < t it holds One easily checks that the integral w.r.t. M is a uniformly integrable martingale and hence has expectation zero. Therefore, using (25), we have for 0 < u < t ≤ T that By Young's inequality, (24) and Lemma 4.6 we get a constant C f such that for any c > 0, Step 2. We now show that In order to estimate the expressions from (28) one can repeat the previous compu- for any c > 0.
For the case v = 0, by using Lipschitz properties of f (which also imply the boundedness of the partial derivatives), we can find a constant C ′ f such that where for some C > 0 Since the sequence (Y n , Z n , U n ) converges in L 2 (W )×L 2 (W )×L 2 (Ñ), condition (A f e) holds, and ∂ y f, ∂ z f, ∂ u f as well as g ′ are bounded and continuous it follows from Vitali's convergence theorem that δ n := E T r e βs κ n (r, s) 2 drds → 0 for n → ∞.
Now we continue with the case v = 0. We first realize that for a given ε > 0 we may choose α > 0 small enough such that This is because from (19), (20) and (15) one gets by a straightforward calculation with L f,g = L f (1 + L g g 1 L 2 (ν) ) where L g is the Litschitz constant of g, and On the set {|v| ≥ α} we use the Lipschitz properties (A f c) and (A f g) to get the estimate This gives for any n ∈ N 2 L 2 (Ñ⊗µ) ds +δ n + ε.
Choosing c in (29) in an appropriate way leads to with C n = C n (α) tending to zero if n → ∞ for any fixed α > 0. We now apply Lemma A.1 and end up with This implies (17). Hence we can take the Malliavin derivative of (14) and get (18) as well as By the same reasoning as for D r,v Y n we may conclude that the RHS of (18) has a càdlàg version which we take for D r,v Y.
(iii) This assertion we get comparing (16) and (18) because of the uniqueness of (Y, Z, U).
(iv) If we consider the pathwise limit lim tցr D r,v Y t of the càdlàg version and compare the RHS of (18) with (33) the assertion follows.

Example: A BSDE related to utility maximization
In [7] a class of BSDEs is considered which appears in exponential utility maximization. For these BSDEs an additional summand arises in the generator which is only locally Lipschitz and is (in the simplest case) of the form: [g α (U s )] ν (see (15)) with g α (x) := e αx − αx − 1 α for some α > 0 and g 1 (x) := 1 for x ∈ R 0 . Consider for 0 ≤ t ≤ T the following BSDE where f g is defined like in (14). Then we have the following assertion: Corollary 4.7. Let ξ ∈ D 1,2 and assume that ξ is a.s. bounded and ν is a bounded measure. If (A f ) is satisfied for f g then the following assertions hold for (34).
with F r,v and Z r,v s,x given in Theorem 4.4 and  Proof. Since ξ is a.s. bounded, the Lévy measure ν is finite and the generator satisfies the conditions of [7, Theorem 3.5.] it follows that Y S∞ < ∞ and |U s (x)| ≤ 2 Y S∞ for P ⊗ λ ⊗ νa.e. (ω, s, x).
From the fact that g α is locally Lipschitz and U is a.e. bounded it follows that (A f ) (especially the Lipschitz condition) can be seen as satisfied also for [g α (U s )] ν : We find a C 1 function g α such that g α = g α on [−2 Y S∞ , 2 Y S∞ ] and Since by (36), g α (U s (x)) = g α (U s (x)), P ⊗ λ ⊗ νa.e., it follows that for all t ∈ [0, T ]

A Appendix
Proof of Lemma 3.3 Step 1. We have the a.s. representation of the Lévy process X as where J t is the jump part of the process according to the Lévy-Itô decomposition (1). We denote B t := γt + σW t . Because of we may restrict ourselves to 'pure jump processes' (i.e. X = J).
Step 2. Assume that X is a compound Poisson process. Then ν (R 0 ) < ∞. The conditional probabilitieŝ P X ∈ Λ N(]0, T ] × R 0 ) = k , k ∈ N, are the distributions of an independent sum of β and the compound Poisson process X, conditioned on the event that the process X jumps k times in ]0, T ]. The probability law of this conditioned compound Poisson process is the same as the law of a piecewise constant process which has exactly k independent, uniformly distributed jumps in [0, T ] whose jump sizes are independently identically distributed according to ν ν(R 0 ) and independent from the jump times. Therefore it holds that T k+1 ν (R 0 ) k+1 ((t 1 ,x 1 ), . . . , (t k ,x k ), (r, v)) : k l=1 x l 1I [t l ,T ] + v1I [r,T ] ∈ Λ = P (X ∈ Λ|N(]0, T ] × R 0 ) = k + 1) = 0, where we used the argument concerning the distribution of a conditioned Poisson process again to come to the last line. Hence, all summands of (37) are zero, which shows the assertion for the special case of this step.
Step 3. To extend the second step to the case of a general pure-jump Lévy process X we split up R 0 into sets S p , p ≥ 1 such that 0 < ν(S p ) < ∞. Without loss of generality set S 1 := {x ∈ R : |x| > 1}. We may assume that the sequence (S p ) p≥1 is infinite, else we would be in the compound Poisson case again. From the proof of the Lévy-Itô decomposition it follows that where the convergence is P-a.s., uniformly in t ∈ [0, T ] and the (X (p) ) given by an notice that X (p) and X (p) are independent. Then From Steps 1 and 2 we conclude that the summands on the RHS of (38) are zero again by which proves Step 3.
Especially, if C n = 0 for all n ∈ N, then g n ≤ 2ε for all n ∈ N.