Backward stochastic Volterra integral equations—Representation of adapted solutions

For backward stochastic Volterra integral equations (BSVIEs, for short), under some mild conditions, the so-called adapted solutions or adapted M-solutions uniquely exist. However, satisfactory regularity of the solutions is difficult to obtain in general. Inspired by the decoupling idea of forward–backward stochastic differential equations, in this paper, for a class of BSVIEs, a representation of adapted M-solutions is established by means of the so-called representation partial differential equations and (forward) stochastic differential equations. Well-posedness of the representation partial differential equations are also proved in certain sense.     

Quadratic backward stochastic differential equations driven by G-Brownian motion: Discrete solutions and approximation

In this paper, we consider backward stochastic differential equations driven by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the $G$-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces.     

Homeomorphism of solutions to backward SDEs and applications

In this paper we study the homeomorphic properties of the solutions to one dimensional backward stochastic differential equations under suitable assumptions, where the terminal values depend on a real parameter. Then, we apply them to the solutions for a class of second order quasilinear parabolic partial differential equations.     

Generalized reflected BSDEs driven by a Lévy process and an obstacle problem for PDIEs with a nonlinear Neumann boundary condition

In this paper, we derive the existence and uniqueness of the solution for a class of generalized reflected backward stochastic differential equations (GRBSDEs in short) driven by a Lévy process, which involve the integral with respect to a continuous process by means of the Snell envelope, the penalization method and the fixed point theorem. In addition, we obtain the comparison theorem for the solutions of the GRBSDEs. As an application, we give a probabilistic formula for the viscosity solution of an obstacle problem for a class of partial differential-integral equations (PDIEs in short) with a nonlinear Neumann boundary condition.     

Solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian Motions

The local existence and uniqueness of the solutions to backward stochastic differential equations(BSDEs, in short) driven by both fractional Brownian motions with Hurst parameter H ∈(1/2, 1) and the underlying standard Brownian motions are studied. The generalization of the It formula involving the fractional and standard Brownian motions is provided. By theory of Malliavin calculus and contraction mapping principle, the local existence and uniqueness of the solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian motions are obtained.     

Backward stochastic differential equations with Azéma's martingale

The aim of this paper is to study backward stochastic differential equations (BSDE) driven by Azéma's martingale and the associated deterministic functional equations. More precisely, we introduce BSDE's vs. Azéma's martingale in a general frame, then we prove that the existence of a solution to a Markovian BSDE implies the existence of a solution to a deterministic functional equation of a new type. Uniqueness for the functional equation is proved in a particular case. Then we discuss BSDE's vs. an asymmetric martingale: half Brownian motion/half Azéma's martingale, which leads to an asymmetric deterministic functional equation.     

Generalized BSDEs and nonlinear Neumann boundary value problems

Summary. We study a new class of backward stochastic differential equations, which involves the integral with respect to a continuous increasing process. This allows us to give a probabilistic formula for solutions of semilinear partial differential equations with Neumann boundary condition, where the boundary condition itself is nonlinear. We consider both parabolic and elliptic equations. Received: 27 September 1996 / In revised form: 1 December 1997     

Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients

We prove a general theorem that the -valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the -valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result.     

Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump

In this paper, we study the existence of martingale solutions of stochastic 3D Navier-Stokes equations with jump, and following Flandoli and Romito (2008) [7] and Goldys et al. (2009) [8], we prove the existence of Markov selections for the martingale solutions.     

General existence results for reflected BSDE and BSDE

In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z. We deal with the case of a bounded terminal condition ξ and a bounded barrier L as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in Essaky and Hassani (submitted for publication) [14]. The work is suggested by the interest the results might have in finance, control and game theory.     

Probabilistic approach to homogenization of a non-divergence form semilinear PDE with non-periodic coefficients

We consider a semilinear partial differential equation (PDE) of non-divergence form perturbed by a small parameter. We then study the asymptotic behavior of Sobolev solutions in the case where the coefficients admit limits in C?esaro sense. Neither periodicity nor ergodicity will be needed for the coefficients. In our situation, the limit (or averaged or effective) coefficients may have discontinuity. Our approach combines both probabilistic and PDEs arguments. The probabilistic one uses the weak convergence of solutions of backward stochastic differential equations (BSDE) in the Jakubowski S-topology, while the PDEs argument consists to built a solution, in a suitable Sobolev space, for the PDE limit. We finally show the existence and uniqueness for the associated averaged BSDE, then we deduce the uniqueness of the limit PDE from the uniqueness of the averaged BSDE.     

On solutions to backward stochastic partial differential equations for Lévy processes

In this paper, we prove the existence and uniqueness of the solution for a class of backward stochastic partial differential equations (BSPDEs, for short) driven by the Teugels martingales associated with a Lévy process satisfying some moment conditions and by an independent Brownian motion. An example is given to illustrate the theory.     

On the well-posedness of coupled forward–backward stochastic differential equations driven by Teugels martingales

We deal with a class of fully coupled forward–backward stochastic differential equations (FBSDEs), driven by Teugels martingales associated with a general Lévy process. Under some assumptions on the derivatives of the coefficients, we prove the existence and uniqueness of a global solution on an arbitrarily large time interval. Moreover, we establish stability and comparison theorems for the solutions of such equations. Note that the present work extends known results proved for FBSDEs driven by a Brownian motion, by using martingale techniques related to jump processes, to overcome the lack of continuity.     

On the orthogonal component of BSDEs in a Markovian setting

In this note we consider a quadratic growth backward stochastic differential equation (BSDE) driven by a continuous martingale M. We prove (in Theorem 3.2) that if M is a strong Markov process and if the BSDE has the form (2.2) with regular data then the unique solution (Y,Z,N) of the BSDE is reduced to (Y,Z), i.e. the orthogonal martingale N is equal to zero, showing that in a Markovian setting the “usual” solution (Y,Z) (of a BSDE with regular data) has not to be completed by a strongly orthogonal component even if M does not enjoy the martingale representation property.     

Infinite horizon BSDEs in infinite dimensions with continuous driver and applications

This paper is devoted to forward-backward systems of stochastic differential equations in which the forward equation is not coupled to the backward one, both equations are infinite dimensional and on the time interval [0, + ∞). The forward equation defines an Ornstein-Uhlenbeck process, the driver of the backward equation has a linear part which is the generator of a strongly continuous, dissipative, compact semigroup, and a nonlinear part which is assumed to be continuous with linear growth. Under the assumption of equivalence of the laws of the solution to the forward equation, we prove the existence of a solution to the backward equation. We apply our results to a stochastic game problem with infinitely many players.     

BSDEs with random default time and related zero-sum stochastic differential games

In this Note we are concerned with backward stochastic differential equations with random default time. The equations are driven by Brownian motion as well as a mutually independent martingale appearing in a defaultable setting. We show that these equations have unique solutions and a comparison theorem for their solutions. As an application, we get a saddle-point strategy for the related zero-sum stochastic differential game problem.     

Anticipated backward stochastic differential equations with non-Lipschitz coefficients

This paper deals with a class of anticipated backward stochastic differential equations. We extend results of Peng and Yang (2009) to the case in which the generator satisfies non-Lipschitz condition. The existence and uniqueness of solutions for anticipated backward stochastic differential equations as well as a comparison theorem are obtained. The existence and uniqueness of L^p(p>2) solutions for anticipated backward stochastic differential equations are also studied.     

Invariant measure for the stochastic Ginzburg Landau equation

The existence of martingale solutions and stationary solutions of stochastic Ginzburg-Landau equations under general hypothesizes on the dimension, the non linear term and the added noise is investigated. With a few more assumptions, it is established that the transition semi-group is well defined and that the stationary martingale solution yields the existence of an invariant measure. Moreover this invariant measure is shown to be unique.     

Stationary backward stochastic differential equations and associated partial differential equations

By replacing the final condition for backward stochastic differential equations (in short: BSDEs) by a stationarity condition on the solution process we introduce a new class of BSDEs. In a natural manner we associate to such BSDEs the periodic solution of second order partial differential equations with periodic structure. Received: 11 October 1996 / Revised version: 15 February 1999     

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.