Given $p \in (1, 2)$, we study $L^p$-solutions of a multi-dimensional backward stochastic differential equation with jumps (BSDEJ) whose generator may not be Lipschitz continuous in $(y,z)-$variables. We show that such a BSDEJ with a p-integrable terminal data admits a unique $L^p$ solution by approximating the monotonic generator by a sequence of Lipschitz generators via convolution with mollifiers and using a stability result.
In this paper, we study conditions under which the solutions of a backward stochastic differential equation with jump remains in a given set of constrains. This property is the so-called "viability property". As an application, we study the comparison theorem for multidimensional BSDEs with jumps.
In this paper we obtain a comparison theorem for backward stochastic partial differential equation (SPDEs) with jumps. We apply it to introduce space-dependent convex risk measures as a model for risk in large systems of interacting components.
In this paper, we study backward stochastic Volterra integral equations introduced in [26, 45] and extend the existence, uniqueness or comparison results for general filtration as in [31] (not only Brownian-Poisson setting). We also consider Lp-data and explore the time regularity of the solution in the It{ô} setting, which is also new in this jump setting.
In this paper{\}we prove the existence of a solution for reflected backward doubly stochastic differential equations with poisson jumps (RBDSDEPs) with one continuous barrier where the generator is continuous and also we study the RBDSDEPs with a linear growth condition and left continuity in $y$ on the generator. By a comparison theorem established here for this type of equation we provide a minimal or a maximal solution to RBDSDEPs.
The aim of this paper is to establish the existence and uniqueness of the solution to a system of nonlinear fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. Our system is Markovian in the sense that initial and terminal values depend on solutions, and are not just fixed random variables. We establish under some monotonicity conditions, the existence and uniqueness of strong solutions of such equations by using a continuation method.
We consider the problem of viscosity solution of integro-partial differential equation(IPDE in short) with one obstacle via the solution of reflected backward stochastic differential equations(RBSDE in short) with jumps. We show existence and uniqueness of a continuous viscosity solution of equation with non local terms, in case the generator is not monotonous and Levy's measure is infinite.
In this paper we study backward stochastic differential equations (BSDEs) driven by the compensated random measure associated to a given pure jump Markov process X on a general state space K. We apply these results to prove well-posedness of a class of nonlinear parabolic differential equations on K, that generalize the Kolmogorov equation of X. Finally we formulate and solve optimal control problems for Markov jump processes, relating the value function and the optimal control law to an appropriate BSDE that also allows to construct probabilistically the unique solution to the Hamilton-Jacobi…
We present a new algorithms to discretize a decoupled forward backward stochastic differential equations driven by pure jump Lévy process (FBSDEL in short). The method is built in two steps. Firstly, we approximate the FBSDEL by a forward backward stochastic differential equations driven by a Brownian motion and Poisson process (FBSDEBP in short), in which we replace the small jumps by a Brownian motion. Then, we prove the convergence of the approximation when the size of small jumps $\eps$ goes to 0. In the second step, we obtain the $L^p$ Hölder continuity of the solution of FBSDEBP and we c…
In this paper, we study a class of multi-dimensional reflected backward stochastic differential equations when the noise is driven by a Brownian motion and an independent Poisson point process, and when the solution is forced to stay in a time-dependent adapted and continuous convex domain ${\cal{D}}=\{D_t, t\in[0,T]\}$. We prove the existence an uniqueness of the solution, and we also show that the solution of such equations may be approximated by backward stochastic differential equations with jumps reflected in appropriately defined discretizations of $\cal{D}$, via a penalization method.
In this paper, we study forward-backward doubly stochastic differential equations driven by Brownian motions and Poisson process (FBDSDEP in short). Both the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs in short) and stochastic Hamiltonian systems arising in stochastic optimal control problems with random jumps are treated with FBDSDEP. Under some monotonicity assumptions, the existence and uniqueness results for measurable solutions of FBDSDEP are established via a method of continuation. Furthermore, the c…
This paper is concerned with the partial information optimal control problem of wa controlled forward-backward stochastic differential equation of jump diffusion with correlated noises between the system and the observation. For this type of partial information optimal control problem, Necessary and sufficient optimality conditions, in the form of Pontryagin maximum principle, for the partial information optimal control are established using a unified way. Moreover, our admissible control process $u(\cdot)$ satisfies the following integrable condition condition: \begin{equation*} \label{eq:1.1…
In this paper, we study backward doubly stochastic differential equations driven by Brownian motions and Poisson process (BDSDEP in short) with non-Lipschitz coefficients on random time interval. The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs in short) is tthe solutionBDSDEP. Under non-Lipschitz conditions, the existence and uniqueness results for measurable solutions of BDSDEP are established via the smoothing technique. Then, the continuous dependence for solutions of BDSDEP is derived. Finally, the prob…
As well-known, transition probabilities of jump Markov processes satisfy Kolmogorov's backward and forward equations. In the seminal 1940 paper, William Feller investigated solutions of Kolmogorov's equations for jump Markov processes. Recently the authors solved the problem studied by Feller and showed that the minimal solution of Kolmogorov's backward and forward equations is the transition probability of the corresponding jump Markov process if the transition rate at each state is bounded. This paper presents more general results. For Kolmogorov's backward equation, the sufficient condition…
The objective of this paper is to give conditions ensuring that the backward partial integro differential equation associated with a multidimensional jump-diffusion with a pure jump component has a unique classical solution; that is the solution is continuous, twice differentiable in the diffusion component and differentiable in time. Our proof uses a probabilistic arguments and extends the results of Pham (1998) to processes with a pure jump component where the jump intensity is modulated by a diffusion process. This result is particularly useful in some applications to pricing and hedging of…
In this work we mainly prove the existence and pathwise uniqueness of solutions to general backward doubly stochastic differential equations with jumps appearing in both forward and backward integral parts. Several comparison theorems under some weak conditions are also given. Finally we apply comparison theorems in proving the existence of solution to some special backward doubly stochastic differential equations with drift coefficient increasing linearly.
We consider a backward stochastic differential equation with jumps (BSDEJ) which is driven by a Brownian motion and a Poisson random measure. We present two candidate-approximations to this BSDEJ and we prove that the solution of each candidate- approximation converges to the solution of the original BSDEJ in a space which we specify. We use this result to investigate in further detail the consequences of the choice of the model to (partial) hedging in incomplete markets in finance. As an application, we consider models in which the small variations in the price dynamics are modeled with a Poi…
We consider the classical problem of maximizing the expected utility of terminal net wealth with a final random liability in a simple jump-diffusion model. In the spirit of Horst et al. (2014) and Santacroce-Trivellato (2014), under suitable conditions the optimal strategy is expressed in implicit form in terms of a forward backward system of equations. Some explicit results are presented for the pure jump model and for exponential utilities.
In the first part of this paper we give a solution for the one-dimensional reflected backward stochastic differential equation (BSDE for short) when the noise is driven by a Brownian motion and an independent Poisson point process. The reflecting process is right continuous with left limits (rcll for short) whose jumps are arbitrary. We first prove existence and uniqueness of the solution for a specific coefficient in using a method based on a combination of penalization and the Snell envelope theory. To show the result in the general framework we use a fixed point argument in an appropriate s…
We consider a class of backward stochastic differential equations (BSDEs) driven by Brownian motion and Poisson random measure, and subject to constraints on the jump component. We prove the existence and uniqueness of the minimal solution for the BSDEs by using a penalization approach. Moreover, we show that under mild conditions the minimal solutions to these constrained BSDEs can be characterized as the unique viscosity solution of quasi-variational inequalities (QVIs), which leads to a probabilistic representation for solutions to QVIs. Such a representation in particular gives a new stoch…
In this paper, we investigate infinite horizon jump-diffusion forward-backward stochastic differential equations under some monotonicity conditions. We establish an existence and uniqueness theorem, two stability results and a comparison theorem for solutions to such kind of equations. Then the theoretical results are applied to study a kind of infinite horizon backward stochastic linear-quadratic optimal control problems, and then differential game problems. The unique optimal controls for the control problems and the unique Nash equilibrium points for the game problems are obtained in closed…
This work deals with backward stochastic differential equation (BSDE) with random marked jumps, and their applications to default risk. We show that these BSDEs are linked with Brownian BSDEs through the decomposition of processes with respect to the progressive enlargement of filtrations. We show that the equations have solutions if the associated Brownian BSDEs have solutions. We also provide a uniqueness theorem for BSDEs with jumps by giving a comparison theorem based on the comparison for Brownian BSDEs. We give in particular some results for quadratic BDSEs. As applications, we study the…
In this paper we investigate a kind of optimal control problem of coupled forward-backward stochastic system with jumps whose cost functional is defined through a coupled forward-backward stochastic differential equation with Brownian motion and Poisson random measure. For this end, we first study the regularity of solutions for this kind of forward-backward stochastic differential equations. We obtain that the value function is a deterministic function and satisfies the dynamic programming principle for this kind of optimal control problem. Moreover, we prove that the value functions is a vis…
In this paper, we prove the existence and uniqueness of the solution of a coupled Mean-Field Forward-Backward SDE system with Jumps. Then, we give an application in the field of storage problem in smart grids, studied in [4] in the case where the production of electricity is not predictable due, for example, to the changes in meteorological forecasts.
We study a problem of utility maximization under model uncertainty with information including jumps. We prove first that the value process of the robust stochastic control problem is described by the solution of a quadratic-exponential backward stochastic differential equation with jumps. Then, we establish a dynamic maximum principle for the optimal control of the maximization problem. The characterization of the optimal model and the optimal control (consumption-investment) is given via a forward-backward system which generalizes the result of Duffie and Skiadas (1994) and El Karoui, Peng an…
We obtain an existence and uniqueness theorem for fully coupled forward-backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the explicit form of the FBSDE solution. Moreover, we embed the associated PIDE into a suitable class of non-local quasilinear parabolic PDEs which allows us to extend the methodology of Ladyzhenskaya et al (1968) to non-local PDEs of this class. Namely, we obtain the existence and uniqueness of a classical solution to both the Cauchy problem and the initial-boundary…
Results on the existence, uniqueness and strict comparison for solutions to a BSDE driven by a multi-dimensional RCLL martingale are established. The goal is to develop a general multi-asset framework encompassing a wide spectrum of nonlinear financial models with jumps, including as particular cases the setups studied by Peng and Xu \cite{PX2009,PX2010} and Dumitrescu et al. \cite{DGQS2018} who dealt with BSDEs driven by a one-dimensional Brownian motion and a purely discontinuous martingale with a single jump.
In this paper, we study a class of stochastic optimal control problem with jumps under partial information. More precisely, the controlled systems are described by a fully coupled nonlinear multi- dimensional forward-backward stochastic differential equation driven by a Poisson random measure and an independent multi-dimensional Brownian motion, and all admissible control processes are required to be adapted to a given subfiltration of the filtration generated by the underlying Poisson random measure and Brownian motion. For this type of partial information stochastic optimal control problem, …
In this paper, we consider the Cauchy problem of semi-linear degenerate backward stochastic partial differential equations (BSPDEs in short) under general settings without technical assumptions on the coefficients. For the solution of semi-linear degenerate BSPDE, we first give a proof for its existence and uniqueness, as well as regularity. Then the connection between semi-linear degenerate BSPDEs and forward backward stochastic differential equations (FBSDEs in short) is established, which can be regarded as an extension of Feynman-Kac formula to non-Markov frame.
Connections between a system of Forward-Backward SDEs and Backward Stochastic PDEs related to the utility maximiza- tion problem is established. Besides, we derive another version of FBSDE of the same problem and prove an existence of a solution
This paper is concerned with the switching game of a one-dimensional backward stochastic differential equation (BSDE). The associated Bellman-Isaacs equation is a system of matrix-valued BSDEs living in a special unbounded convex domain with reflection on the boundary along an oblique direction. In this paper, we show the existence of an adapted solution to this system of BSDEs with oblique reflection by the penalization method, the monotone convergence, and the a priori estimates.
In this paper, we study a Backward Stochastic Differential Equation with Jumps (BSDEJs in short) where the jumps have infinite activity. Following a forward approach based on Exponential Quadratic semimartingale, we prove the existence of solution of Quadratic BSDEJs with unbounded terminal condition and quadratic growth in z.
We define fully coupled forward-backward stochastic differential equations on spaces related to continuous time, finite state Markov Chains. Existence and uniqueness results of the fully coupled forward-backward stochastic differential equations on Markov Chains are obtained.
This paper extends the idea of E.Gobet, J.P.Lemor and X.Warin from the setting of Backward Stochastic Differential Equations to that of Backward Doubly Stochastic Differential equations. We propose some numerical approximation scheme of these equations introduced by E.Pardoux and S.Peng.
In this paper, by virtue of Malliavin calculus, we establish a relationship between backward doubly stochastic differential equations with random coefficients and quasilinear stochastic PDEs, and thus extend the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [14] to non-Markovian case.
In this paper, we study backward doubly stochastic integral equations of the Volterra type (BDSIEVs in short). Under uniform Lipschitz assumptions, we establish an existence and uniqueness result.
In this paper, we consider the fully coupled forward-backward stochastic functional differential equations (FBSFDEs) with stochastic functional differential equations as the forward equations and the generalized anticipated backward stochastic differential equations as the backward equations. We will prove the existence and uniqueness theorem for FBSFDEs. As an application, we deal with a quadratic optimal control problem for functional stochastic systems, and get the explicit form of the optimal control by virtue of FBSFDEs.
This paper is devoted to study the asymptotic properties for the solution of decoupled forward backward stochastic differential equations with delayed generator. As an application, we establish a large deviation principe for solution of the backward equation.
We prove an existence and uniqueness result for generalized backward doubly stochastic differential equations driven by Lévy processes with non-Lipschitz assumptions.
In this paper, we study backward stochastic differential equations (BSDEs shortly) with jumps that have Lipschitz generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. Under just integrability on the data we show that such equations admits a unique solution which belongs to class D.
We study reflected solutions of one-dimensional backward doubly stochastic differential equations (BDSDEs in short). The "reflected" keeps the solution above a given stochastic process. We get the uniqueness and existence by penalization. For the existence of backward stochastic integral, our proof is different from [KKPPQ] slightly. We also obtain a comparison theorem for reflected BDSDEs.
We prove the existence of the unique solution of a general Backward Stochastic Differential Equation with quadratic growth driven by martingales. Some kind of comparison theorem is also proved.
We study the well solvability of nonlinear backward stochastic evolutionary equations driven by a space-time white noise. We first establish a novel a priori estimate for solution of linear backward stochastic evolutionary equations, and then give an existence and uniqueness result for nonlinear backward stochastic evolutionary equations. A dual argument plays a crucial role in the proof of these results. Finally, an example is given to illustrate the existence and uniqueness result.
Existence and uniqueness results of fully coupled forward stochastic differential equations without drifts and backward stochastic differential equations in a degenerate case are obtained for an arbitrarily large time duration.
The connection between forward backward doubly stochastic differential equations and the optimal filtering problem is established without using the Zakai's equation. The solutions of forward backward doubly stochastic differential equations are expressed in terms of conditional law of a partially observed Markov diffusion process. It then follows that the adjoint time-inverse forward backward doubly stochastic differential equations governs the evolution of the unnormalized filtering density in the optimal filtering problem.
This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is…
We extend the Yamada-Watanabe condition for pathwise uniqueness to stochastic differential equations with jumps, in the special case where small jumps are summable.
Under quasi-monotone assumptions for coefficients, we show one kind of comparison theorem for multi-dimensional\textbf{\}backward doubly stochastic differential equations on infinite horizon. An example is given as well.
In this paper, a class of non-Markovian forward-backward doubly stochastic systems is studied. By using the technique of functional Itô (or path-dependent) calculus, the relationship between the systems and related path-dependent quasi-linear stochastic partial differential equations (SPDEs in short) is established, and the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98 (1994), pp. 209--227] is developed to the non-Markovian situation. Moreover, we obtai…
In this paper, a class of generalized backward doubly stochastic differential equations whose coefficient contains the subdifferential operators of two convex functions (also called generalized backward doubly stochastic variational inequalities) are considered. By means of a penalization argument based on Yosida approximation, we establish the existence and uniqueness of the solution. As an application, this result is used to derive existence result of stochastic viscosity solution for a class of multivalued stochastic Dirichlet-Neumann problems.
