Feynman–Kac formula for BSDEs with jumps and time delayed generators associated to path-dependent nonlinear Kolmogorov equations

We consider a system of forward backward stochastic differential equations (FBSDEs) with a time-delayed generator driven by Lévy-type noise. We establish a non-linear Feynman–Kac representation formula associating the solution given by the FBSDEs system to the solution of a path dependent nonlinear Kolmogorov equation with both delay and jumps. Obtained results are then applied to study a generalization of the so-called large investor problem, where the stock price evolves according to a jump-diffusion dynamic.


Introduction
Stochastic delay differential equations are derived as a natural generalisation of stochastic ordinary differential equations by allowing coefficients' evolution to depend not only on the present state, but also on past values.
In this article we analyze a stochastic process described by a system of Forward-Backward Stochastic Differential Equations (FBSDEs).The forward path-dependent equation is driven by a Lévy process, while the backward one presents a path-dependent behaviour with dependence on small delay δ.
Also, for a fixed delay δ > 0 we set We emphasize that the study of a path-dependent Kolmogorov equation whose generator f depends on both a delayed term (u(•, φ)) t and on a jump operator J u(t, φ), represents the main novelty we provide in this paper.
Under appropriate assumptions on the coefficients, the deterministic non-anticipative functional u : [0, T ] × Λ → R given by the representation formula is a mild solution of the Kolmorogov Equation (1.1).
Concerning the stochastic process Y t,φ (t) in Eq. (1.5), provided conditions in Assumptions 2.1 -3.1, are fulfilled, we know that the quadruple X t,φ , Y t,φ , Z t,φ , U t,φ s∈[t,T ] , is the unique solution of the system of FBSDEs on [t, T ] given by X t,φ (s) = φ(t) + s t b(r, X t,φ )dr + s t σ(r, X t,φ )dW (r) γ(r, X t,φ , z) Ñ (dr, dz) Y t,φ (s) = h(X T,φ ) + T s f r, X t,φ , Y t,φ (r), Z t,φ (r), Ũ t,φ (r), Y t,φ r dr where W stands for a l-dimensional standard Brownian motion.We assume a delay δ ∈ R + .Consequently, the notation Y t,φ r , appearing in the generator f of the backward dynamic in the system (1.6), stands for the delayed path of the process Y t,φ restricted to [r − δ, r], namely Y t,φ r := Y t,φ (r + θ) θ∈[−δ,0] . (1.7) In Eq. (1.6), Ñ models a compensated Poisson random measure, independent from W , with associated Lévy measure ν.This stochastic term appears also in the definition of the integral term Ũ : [0, T ] → R related to the jump process by we need to introduce to express the solution of (1.1) via the FBSDE system (1.6).
The connection between probability theory and PDEs is a widely analysed subject, the well-known Feynman-Kac formula ( [27]) being one of its main turning points stating that solutions for a large class of second order PDEs of both elliptic and parabolic type, can be expressed in terms of expectation of a diffusion process.Latter result has been then generalised by Pardoux and Peng in [32,33] to show the connection between Backward Stochastic Differential Equations (BSDEs) and a system of semi-linear PDEs, then proving the non linear Feynman-Kac formula within the Markovian setting.Concerning the non-Markovian scenario, we know from [35,38] that a non linear Feynman-Kac formula can be still established, associating a path-dependent PDE to a non-Markovian BSDE.More recently, the introduction of horizontal and vertical derivatives of non-anticipative functionals on path spaces by Dupire [19], Cont [7] and Fournié [8], facilitated the formulation of a new class of path dependent PDEs and the introduction of the so-called viscosity solution concept, see [22], [23], [38], for more details.For a complete overview about stochastic calculus with delay we refer to [30], [31] and [43].
A connection between time-delayed BSDEs and path-dependent PDEs has been proved by an infinite dimensional lifting approach in [24] and [29].In [3], the authors considers a BSDE driven by a Brownian motion and a Poisson random measure that provides a viscosity solution of a system of parabolic integral-partial differential equations.In [11], the existence of a viscosity solution to a path dependent nonlinear Kolmogorov equation (without jumps) and the corresponding nonlinear Feynman-Kac representation has been proved.
In this paper we deal with the notion of mild solution which can be seen as intermediate notion for solution of a PDE lying in between the notions of classical and viscosity solutions.In [26], the authors provide the definition of mild solution for non linear Kolmogorov equations along with its link with a specific stochastic process.The latter has been also proved for Semilinear Parabolic Equations in [25], where the definition of the generalized directional gradient is firstly introduced.The concept of mild solution together with the generalized directional gradient to handle pathdependent Kolmogorov equation with jumps and delay has been widely analyzed in the functional formulation, see, e.g., [13].Moreover, a discrete-time approximation for solutions of a system of decoupled FBSDEs with jumps have been proved in [6] by means of Malliavin calculus tools.
Concerning the theory of BSDE with a dependence on a delay, in [17], the authors proved the existence of a solution for a BSDE with a time-delayed generator that depends on the past values of the solution.In particular, both existence and uniqueness are proved assuming a sufficiently small time horizon T or a sufficiently small Lipschitz constant for the generator.Let us underline that the latter has an equivalent within our setting, as we state in Remark 3.1.Moreover, in [15,18] the authors defined a path-dependent BSDE with time delayed generators driven by Brownian motions and Poisson random measures, with coefficients depending on the whole solution's path.
In [2], following a different approach, namely considering systems with memory and jumps, the authors provide a characterization of a strong solution for a delayed SDE with jumps, considering both L p -type space and càdlàg processes to provide a non-linear Feynman-Kac representation theorem.
The present paper is structured as follows: we start providing notations and problem setting in Section 2, according to the theoretical framework developed by [15] and [17]; in Section 3 we study the well-posedness of the path-dependent BSDE mentioned appearing in the Markovian FBSDEs stystem (1.6) following the approach in [11] by additionally considering jumps; in Section 4 we provide a Feynman-Kac formula relating the BSDE to the Kolmogorov Equation defined in (1.1) to then generalise results in [13] by considering a dependence in the generator f of the backward dynamic on a delayed L 2 term, namely Y t,φ r , for a small delay δ; in Section 5 we derive the existence of a mild solution for the Kolmogorov Equation within the setting developed in [25]; finally, in Section 6, we provide an application based on the analyzed theoretical setting, i.e. a version of the Large Investor Problem characterised by a jump-diffusion dynamic.
On a probability space Ω, F , P , we consider a standard l-dimensional Brownian motion W and a homogeneous Poisson random measure N on R + ×(R\{0}), independent from W , with intensity ν.With the notation R 0 := R\{0}, we also define the compensated Poisson random measure Ñ defined on R + × R 0 by Ñ (dt, dz) := N (dt, dz) − ν(dz)dt . (2.1) For the sake of completeness, we recall that the term ν(dz)dt is known as the compensator of the random measure N .Moreover, we assume that the Lèvy measure ν satisfies being a standard assumption when dealing with financial applications.The reader could see, e.g., in [4] further details about the stochastic integration with the presence of jumps.

The forward-backward delayed system
In this section we introduce the delayed forward-backward system, assuming path dependent coefficients for the forward and the backward components and a dependence on a small delay into the generator f and the presence of jumps modeled via a compound Poisson measure.Furthermore, the equation is formulated on a general initial time t and initial values.Thus, we need to equip the backward equation with a suitable condition in [0, t), as we introduced in Equation (2.6).
On previously defined probability space, we consider a filtration F t = {F t s } s∈[0,T ] , which is nothing but the one jointly generated by W (s ∧ T ) − W (t) and N (s ∧ T, •) − N (t, •), augmented by all P-null sets.We emphasize that F t depends explicitly on t, namely the arbitrary initial time in [0, T ] for the dynamic in Eq. (1.6).
Furthermore,the components of the solution of the backward dynamic are defined in the following Banach spaces: • S 2 t (R) denotes the space of (equivalence class of) F t -adapted, product measurable càdlàg processes Y : The spaces S 2 t (R), H 2 t (R l ) and H 2 t,ν (R) are endowed with the following norms: for some β > 0, to be precised later.
The main goal is to find a family of stochastic processes X t,φ , Y t,φ , Z t,φ , U t,φ t,φ∈[0,T ]×Λ adapted to F t such that the following decoupled forward-backward system holds a.s. (2. 3) It is worth to mention that, differently from [15], we work in a Markovian setting, enforcing an initial condition over all the interval [0, t].More precisely, for both forward and backward equations, the values of the solution X t,φ need to be known in the time interval [0, t].Analogously, regarding the backward component, the values of Y t,φ , Z t,φ and U t,φ need also to be prescribed for s ∈ [0, t].
Remark 2.1.The δ-delayed feature concerns only Y , but we emphasize that it is possible to generalize this result to treat the case where both Z and U depend on their past values for a fixed delay δ.
For the sake of simplicity, we consider the case with Y r , hence limiting ourselves to just one, the process Y , L 2 delayed term.As a consequence, the latter implies that we will have to consider a larger functional space to properly define the contraction that is an essential step to prove the fix point argument in Th. 3.2.

The forward path-dependent SDE with jumps
We first study the forward component of X appearing in the system (2.3).It is defined according to the following equation: (2.4) The existence and the pathwise uniqueness for a solution of (2.4) is a classical result, proved, for instance, in [42], exploiting a Picard iteration approach.We assume the following assumptions to hold: (A 1 ) b, σ and γ are continuous; (A 2 ) there exists ℓ > 0 such that for any t ∈ [0, T ], φ, φ ′ ∈ Λ; If (A 1 ), (A 2 ), (A 3 ) hold, then there exists a solution to (2.4) and this solution is pathwise unique, see, e.g., [3] for a detailed proof.

The backward delayed path-dependent SDE with jumps
We now focus on the BSDE appearing in the system (2.3), namely (2.5) for a finite time horizon T < ∞ and φ ∈ Λ := D [0, T ]; R d .The path-dependent process X t,φ represents the solution of the forward SDE with jumps of Eq. (2.4), while Ñ models the compensated Poisson random measure described in Eq. (2.1) and W is a l-dimensional Brownian motion.
We recall that, when we fix the delay term δ, the notation Y t,φ r stands for the path of the process restricted to [r − δ, r], according to Eq.(1.7).Notice that the terminal condition enforced by h depends on the solution of the forward SDE (2.4) as well as the solution (Y, Z, U ) of the backward component considered in the time interval [t, T ].
Differently from the framework studied by Delong in [15], we consider a general initial time s ∈ [0, t).As highlighted in [11], the Feynman Kac formula would fail with standard prolongation.
Thus, an additional initial condition has to be satisfied over the interval [0, t], given by We remark that the supplementary initial condition stated in Eq. (2.6) represents one of the main difference between Theorem 3.2 and Theorem 14.1.1 in [15].

The Well Posedness of the BSDE
Concerning the delayed backward SDE (2.5), we will assume the following to hold.
Remark 3.1.In order to show both existence and uniqueness of a solution to the backward part of the system (1.6) and to obtain the continuity of Y t,φ with respect to φ, we need to impose K or δ to be small enough.More precisely, we will assume that there exists a constant χ ∈ (0, 1), such that: The main difference between our result and Theorem 3.4 in [11] relies in the presence of a jump component in the dynamics of the unknown process Y t,φ : this further term implies a stronger bound in the condition enforced in Eq. (3.1).
Hence, if K or δ are be small enough to satisfy the condition stated in Eq. (3.1), then there exists a unique solution of (2.5) and the following theorem holds Theorem 3.2.Let assumptions A 4 , A 5 , A 6 hold.If condition (3.1) is satisfied, then there exists a unique solution The proof of Th. 3.2 is provided in Appendix 7 and it is mainly based on the Banach fixed point theorem.
We emphasize that similar results hold also for multi-valued processes, namely or the necessity of introducing the n-fold iterated stochastic integral, see [5], [6] or [13] (Sec.2.1), for further details.

The Feynman-Kac formula
In what follows we prove that the solution of of Eq. (1.6), namely the path-dependent forward-backward system with delayed generator f and driven by a Lévy process, can be connected to solution of path-dependent PIDE represented by the non-linear Kolomogorov equation (1.1).

The Delfour-Mitter space
According to [2], we need the solution of the forward SDE (1.6) to be a Markov process as to derive the Feynman-Kac formula.The Markov property of the solution is fully known for the SDE without jumps, i.e. when γ = 0, see [30] (Th.III.1.1).Moreover, the Markov property also holds, by enlarging the state space, for the solution in a setting analogous to that of Eq. (2.4), see [13] (Prop.2.6) where driving noises with independent increments are considered.Since X t,φ : [0, T ] × D([0, T ]; R d ) → R d in not Markovian, we enlarge the state space by considering the process X as a process of the path, by introducing a suitable Hilbert space, as described in [24] and [29], where they present a product-space reformulation of (4.1) splitting the present state X(t) from the past trajectory X [0,t) by a particular choice of the state space.
Accordingly, we enlarge the state space of our interest, starting from paths defined on the Skorohod space D [0, T ]; R d to then consider a new functional space, the so-called Delfour-Mitter space as in-depth analyzed in, e.g., [2] and [13].Moreover, we rewrite the second one as a function on [−t, 0) via a change of variable (in time) and then we lengthen it towards the past up to [−T, 0) to consider the whole path.The latter approach allows us to identify the càdlàg process X t,φ with the enlarged process X t,η,x defined over M 2 , namely Here, X(s) denotes the present state that is well defined for any time s ∈ [0, T ], X s+r denoting the segment of the path described by X, which takes values in It is worth mentioning that M 2 has a Hilbert space structure, endowed with the following scalar product By the continuous injections D ⊂ M 2 , we can formulate the forward equation with (2.4) and f are the given coefficients of Eq. (2.4).
We have to additionally impose that b, σ, γ, f and h are locally Lipschitz-continuous with respect to φ ∈ Λ in the L 2 -norm: as understood in [25], the function u in Eq. (4.5), namely the solution of the Kolmogorov Equation (1.1), is locally Lipschitz in x with polynomial growth.Thus, in order to have the same regularity for the solution of the BSDE system with forward Eq. (4.1), we require that the coefficients b, σ and γ to be locally Lipschitz.
Assumption 4.1.There exists K ≥ 0 and m ≥ 0 such that: Within this setting, lifting the state space turns out to be particularly convenient to investigate differentiability properties of the solution and to relate the solution of Eq. (4.1) (combined with a proper backward equation) to the solutions of the non-linear Kolmogorov equation defined by Eq. (1.1), defined on [0, T ] × Λ.
Remark 4.1.It is also possible to work directly on the Skorohod space D. However, since D is not a separable Banach space, one has to consider weaker topologies on D, following a semi-group approach like the one developed by Peszat and Zabczyk in [39].

Main theorem
The main result of this section provides a nonlinear version of the Feynman-Kac formula in the case where the process X t,φ has jumps and the generator of the backward dynamic f depends on the past values of Y .
for any (t, φ) ∈ [0, T ] × Λ , where Y t,φ is the solution of the following BSDE ) Moreover, the solution of the forward backward equation To prove the representation formula (4.7), we adapt the proof of Theorem 4.10 of [11] by adding the contribution of U and Ũ , respectively modelling the process and the integral term connected to the jump component.
Proof.We follow the Picard iteration scheme, hence considering the iterative process of the BSDE with delayed generator driven by Lévy process described by with Y 0,t,φ ≡ 0, Z 0,t,φ ≡ 0 and U 0,t,φ ≡ 0.
Let us suppose that there exists a F-progressively measurable functional u n : [0, T ] × Λ → R such that u n is locally Lipschitz and Y n,t,φ (s) = u n (s, X t,φ ) for every t, s ∈ [0, T ] and φ ∈ Λ.
According to Theorem 4.10 in [11], we consider the delayed term the delayed term reads and our equation becomes We fix n and we define ψ where h : Since u n is locally Lipschitz-continuous, one can show that ψ is also locally Lipschitz in ϕ.

Mild solution of the Kolmogorov Equation
In this section, we prove the existence of mild solution of the Path-dependent Partial Integro-Differential Equation (PPIDE) Kolmorogov equation (1.1) showing a dependence both on a delayed term and on integral term modelling jumps.For the sake of completeness, let us start recalling the following notion of mild solution.
Definition 5.1.A function u : [0, T ] × M 2 → R is a mild solution to Eq. (1.1) if there exists C > 0 and m ≥ 0 such that for any t ∈ [0, T ] and any φ 1 , φ 2 ∈ M 2 , u satisfies and the following equality holds true for all t ∈ [0, T ] and φ ∈ M 2 , (u(•, φ)) s ) being the delayed term defined in Eq. (1.4).(P t,s ) 0≤t≤s≤T corresponds the Markov transition semigroup corresponding to the operator L introduced in Eq. (1.2), that in the lifted setting corresponds to the generator of the Markov semigroup associated to forward equation appearing in Eq. (4.1), for more details, see, e.g., [25]).
The next theorem represents the core result of this section.
where ϕ t is defined according to Eq. (4.3) and u coincides with Eq. (4.7).
Proof.Let us consider the backward component of the FBSDE described in Eq. (1.6) for s ∈ By Theorem 3.2, there exists a representation formula for the solution of the BSDE (5.4).Moreover, by means of Eq. (4.7), we can write the delayed term Y r as a function of the path of solution of the forward dynamic X t,φ and, thus, we obtain ũ(t, φ) that we can plug into (5.4),leading to .
At this point, we enlarge the state space going through M 2 coefficients analogously to the proof of Theorem 3.2, obtaining Y t,η,x (s) = h(X s,η,x T , X s,η,x (T )) + T s ψ r, X t,η,x r , X t,η,x (r), Y t,η,x (r), Z t,η,x (r), Ũ t,η,x (r), dr U t,η,x (r, z) Ñ (dr, dz) , (5.5) recalling the map ψ defined in Eq. ( 4.8), we may obtain a BSDE in the same setting of [13].Taking the expectation and exploiting the representation formulas described by Theorem 4.5 in [13], we know that there exists a process and we conclude identifying u with the restriction of v on the set of càdlàg paths Λ.
Remark 5.2 (Uniqueness).Let us underline that the aforementioned lack of uniqueness can be overcome by considering the problem of uniqueness of the mild solution in the case of f being independent of the term (u(•, φ)) t , with passages similar to the proof of 4.2.Indeed, let us take two mild solutions u 1 and u 2 of the path-dependent PDE (1.1).We define Using these drivers we can consider the following BSDEs: for which there exist unique solutions Y i,t,φ , Z i,t,φ , U i,t,φ ∈ S 2 t (R) × H 2 t (R l ) × H 2 t,ν (R) for i = 1, 2 .By Theorem 4.2 we see that Y i,t,φ (s) = v i (s, X t,φ ), for all s ∈ [0, T ] , a.s., for any (t, φ) ∈ [0, T ] × Λ, where v i : [0, T ] × Λ → R is (the restriction) of the solution of (4.7), i.e.
Hence, by Theorem 5.1, we obtain that the functions v i are solutions of the PDE of type (1.1), but without the delayed terms v i (•, φ) t : Since u i is also solution to equation (1.1), by using Theorem 4.5 from [13] we know that By asking that f satisfies assumption (A 8 ), we know that BSDEs (5.6) become a single equation, with i = 1, 2 , for which we have uniqueness from Theorem 3.2.

Financial Application
In this section we provide a financial application moving from the model studied in, e.g.[12], or [20].We consider a generalization of the so-called Large Investor Problem, where a so called large investor wishes to invest on a given market, buying or selling a stock.The investor has the peculiarity that his actions on the market can affect the stock price.We further generalize previous results in [11] [20] or by asking, in addition to a small time delay between the action of the large investor and the reaction of the market, the dynamic of the risky asset to be driven by a Poisson random measure.
More specifically, let us denote the investor's strategy by π and the investment portfolio by X π .Then, according to the theory previously developed, here X π is modeled as a càdlàg process and its past X π r may affects directly the drift µ of the stock rate.Consequently we consider the following dynamics where r, µ, σ and γ are F W, Ñ -predictable processes, being F W, Ñ the natural filtration associated to the Brownian motion W , that are required to be adapted to the Poisson random measure Ñ , with compensator is defined according to Eq. (2.1).The initial datum s 0 belongs to the class of càdlàg stochastic process with values in R.
The total amount of the portfolio of the large investor is described by where X π (t) and π(t) the present state of the processes with values in R.
The goal is to find an admissible replicating strategy π ∈ A for a claim F (S(T )).We have that the portfolio X evolves according to , where, at any time t ∈ [0, T ] by π(t), represents the amount invested in the risky asset S, while by X(t) − π(t), is the amount invested in the riskless bond S 0 , and X π (T ) = h (S) represents the final condition.
Hence, for t ∈ [0, T ], we have where, for the sake of readability, we denoted: We then impose that functions r, µ, σ and γ are such that the function Furthermore, we introduce a simplification to decouple the BSDE (6.2) from the stock forward dynamic (6.1), as to fit the setting of Theorem 3.2.Indeed, instead of having h(S) depending on the whole path of S, we consider the terminal function h that explicitly depends only on W , Ñ , Z and U .Thus, the functional h encodes the information at terminal time T of the forward path of S and it reads h (W, X, Z, U ) := h(W, X, σ −1 (•) , U ) , while satisfying a Lipschitz condition: We underline that assuming suitable conditions on µ, σ and γ, e.g.asking µ bounded and σ, γ constant, then h satisfies (A 6 ) and the condition in Eq. (6.4).
Therefore we can rewrite (6.2) as and we deduce from Theorem 3.2 that, under proper assumptions on the coefficients, there exists a unique solution (X, Z, U ) to equation (6.5).Consequently, (6.2) admits a unique solution (X, π), where In order to obtain the connection with the associated PDE we first consider the decoupled forward-backward stochastic system: where the BSDE coefficients are defined according to (6.3).
If Assumptions 3.1 hold for the coefficients, we can exploit 3.2 to derive that there exists a unique solution X t,φ , π t,φ (t,φ)∈[0,T ]×Λ of the system (6.6).By Theorem 4.2, for the solution (X, Z, U ) of the backward component in the above system, we can write that for every where, exploiting Theorem 5.1, u (t, φ) = X t,φ (t) is a specific restriction of the mild solution, according to Eq. ( 5.3), of the following path-dependent PDE:

Conclusions and Future Development
The core result of this paper relies in deriving a stochastic representation for the solutions of a non-linear PDE and associating the solution of the PDE to a FBSDE with jumps and time-delayed generator.The presence of jumps both in the forward and in the backward dynamic and, moreover, the dependence of the generator on a time-delayed coefficient represents the main aspect of novelty arising in the analysis of the FBSDE system.Furthermore, we present an application for a large investor problem admitting a jump diffusion dynamic.Throughout the article we mention some possibilities to generalize the setting of our equations such as considering a dependence of f also on a delayed term for the processes Z and U (see Remark 2.1 for more details) or in-depth analyzing the choice of a weaker topology (Remark 4.1).Another possible modelling choice deals with considering a further delay term affecting the forward process, see [28] for more details.Furthermore, it might deserve attention to investigate a discretization scheme for this equation, recalling the work in [6], for the considered equations as to obtain a numerical algorithm based on Neural Networks methods to efficiently compute an approximated solution for the considered FBSDE.

Proof of Theorem 3.2
Proof.The existence and the uniqueness are obtained by the Banach fixed point theorem.We consider φ fixed in Λ and we define the map Γ on A with A := C [0, T ] ; S 2 0 (R) .For R ∈ A, we define Γ(R) := Y , where, for t ∈ [0, T ], the triple of adapted processes Y t (s), Z t (s), U t (s, z) s∈[t,T ] is the unique solution of the following BSDE For s ∈ [0, t] we prolong the solution by taking Y t (s) := Y s (s) and Z t (s) = U t,φ (s) := 0.
Step 1.Let us first show that Γ takes values in the Banach spaces A. We take R ∈ A and we will prove that Y := Γ(R) ∈ A. Thus, for every t ∈ [0, T ] we have to show that and that the application Let t ∈ [0, T ] be fixed and t ′ ∈ [0, T ]; with no loss of generality, we will suppose that t < t ′ and t ′ − t < δ.
Concerning the solution of the BSDE defined in (.1), we obtain the following estimate We start by proving that as t ′ → t.By plugging the explicit solution and applying Doob's inequality, we get From the absolute continuity of the Lebesgue integral, we deduce that us denote for short, only throughout this step, We apply Itô's formula to e βs |∆Y (s) | We note that the following estimate  We now choose β, a > 0 such that hence we obtain where Exploiting thus assumptions (A 3 ) and (A 5 ) together with the fact that X •,φ is continuous and bounded, we have Since R ∈ A, and therefore we have as t ′ → t.
We are left to show that the term Since the map t → Y t (t) is deterministic, we have from equation (.1), and therefore we obtain Taking again into account the fact that R ∈ A, previous step and assumptions (A 3 ) and (A 5 ), we infer that Concerning the term Analogously, we can infer that Step II.
Step 2. We are going to prove that Γ is a contraction on A with respect to the norm Let us recall that Γ : A → A is defined by Γ (R) = Y being Y the process coming from the solution of the BSDE (.1).

Let us
For the sake of brevity, we will denote in what follows Eventually, since R is chosen arbitrarily, it follows that the application Γ is a contraction on A. Therefore, there exists a unique fixed point Γ(R) = Y ∈ A and this finishes the proof of the existence and a uniqueness of a solution to BSDE with delay and driven by Lèvy process, described by Eq. (2.5).