We consider the simulation of a system of decoupled forward-backward stochastic differential equations (FBSDEs) driven by a pure jump Lévy process $L$ and an independent Brownian motion $B$. We allow the Lévy process $L$ to have an infinite jump activity. Therefore, it is necessary for the simulation to employ a finite approximation of its Lévy measure. We use the generalized shot noise series representation method by Rosinski (2001) to approximate the driving Lévy process $L$. We compute the $L^p$ error, $p\ge2$, between the true and the approximated FBSDEs which arises from the finite truncation of the shot noise series (given sufficient conditions for existence and uniqueness of the FBSDE). We also derive the $L^p$ error between the true solution and the discretization of the approximated FBSDE using an appropriate backward Euler scheme.
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
The theory of backward SDEs extends the predictable representation property of Brownian motion to the nonlinear framework, thus providing a path-dependent analog of fully nonlinear parabolic PDEs. In this paper, we consider backward SDEs, their reflected version, and their second-order extension, in the context where the final data and the generator satisfy $L^1$-type of integrability condition. Our main objective is to provide the corresponding existence and uniqueness results for general Lipschitz generators. The uniqueness holds in the so-called Doob class of processes, simultaneously under…
This paper deals with generalized backward doubly stochastic differential equations driven by a Lévy process (GBDSDEL, in short). Under left or right continuous and linear growth conditions, we prove the existence of minimal (resp. maximal) solutions.
In this paper, we prove the existence and uniqueness result of the reflected BSDE with two continuous barriers under monotonicity and general increasing condition on $y$, with Lipschitz condition on $z$.
The work concerns a type of backward multivalued McKean-Vlasov stochastic differential equations. First, we prove the existence and uniqueness of solutions for backward multivalued McKean-Vlasov stochastic differential equations. Then, it is presented that their solutions depend continuously on the terminal values. Finally, we give a probabilistic interpretation for viscosity solutions of nonlocal quasi-linear parabolic variational inequalities.
This paper is intended to give a probabilistic representation for stochastic viscosity solution of semi-linear reflected stochastic partial differential equations with nonlinear Neumann boundary condition. We use it connection with reflected generalized backward doubly stochastic differential equation.
In this paper we study the homeomorphic properties of the solutions to one dimensional backward doubly 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 stochastic partial differential equations.
This paper is intended to give a representation for stochastic viscosity solution of semi-linear reflected stochastic partial differential equations with nonlinear Neumann boundary condition. We use its connection with reflected generalized backward doubly stochastic differential equations.
In this paper we present two numerical schemes of approximating solutions of backward doubly stochastic differential equations (BDSDEs for short). We give a method to discretize a BDSDE. And we also give the proof of the convergence of these two kinds of solutions for BDSDEs respectively. We give a sample of computation of BDSDEs.
We provide an existence and uniqueness theory for an extension of backward SDEs to the second order. While standard Backward SDEs are naturally connected to semilinear PDEs, our second order extension is connected to fully nonlinear PDEs, as suggested by Cheridito et.al. In particular, we provide a fully nonlinear extension of the Feynman-Kac formula. Unlike the earlier papers, the alternative formulation of this paper insists that the equation must hold under a non-dominated family of mutually singular probability measures. The key argument is a stochastic representation, suggested by the opt…
We deal with reflected solutions of anticipated backward doubly stochastic differential equations (RABDSDEs) driven by Teugels martingales associated with Lévy process under a Lipschitz generator where the coefficients of these BDSDEs depend on the future and present value of the solution $\left( Y,Z\right) $. Also we study the existence of a solution for anticipated BDSDEs.
In this paper, a class of reflected generalized backward doubly stochastic differential equations (reflected GBDSDEs in short) driven by Teugels martingales associated with Lévy process and the integral with respect to an adapted continuous increasing process is investigated. We obtain the existence and uniqueness of solutions to these equations. A probabilistic interpretation for solutions to a class of reflected stochastic partial differential integral equations (PDIEs in short) with a nonlinear Neumann boundary condition is given.
In this paper we study a zero-sum switching game and its verification theorems expressed in terms of either a system of Reflected Backward Stochastic Differential Equations (RBSDEs in short) with bilateral interconnected obstacles or a system of parabolic partial differential equations (PDEs in short) with bilateral interconnected obstacles as well. We show that each one of the systems has a unique solution. Then we show that the game has a value.
This paper investigates a numerical probabilistic method for the solution of some semilinear stochastic partial differential equations (SPDEs in short). The numerical scheme is based on discrete time approximation for solutions of systems of decoupled forward-backward doubly stochastic differential equations. Under standard assumptions on the parameters, the convergence and the rate of convergence of the numerical scheme is proven. The proof is based on a generalization of the result on the path regularity of the backward equation.
We study optimal stochastic control problem for non-Markovian stochastic differential equations (SDEs) where the drift, diffusion coefficients, and gain functionals are path-dependent, and importantly we do not make any ellipticity assumption on the SDE. We develop a controls randomization approach, and prove that the value function can be reformulated under a family of dominated measures on an enlarged filtered probability space. This value function is then characterized by a backward SDE with nonpositive jumps under a single probability measure, which can be viewed as a path-dependent versio…
In this note, we consider Jensen's inequality for the nonlinear expectation associated with backward SDEs driven by $G$-Brownian motion ($G$-BSDEs for short). At first, we give a necessary and sufficient condition for $G$-BSDEs under which one-dimensional Jensen inequality holds. Second, we prove that for $n>1$, the $n$-dimensional Jensen inequality holds for any nonlinear expectation if and only if the nonlinear expectation is linear, which is essentially due to Jia (Arch. Math. 94 (2010), 489-499). As a consequence, we give a necessary and sufficient condition for $G$-BSDEs under which the $…
In this paper, we consider quadratic forward-backward SDEs (QFBSDEs), for {which} the drift in the forward equation does not satisfy the standard globally Lipschitz condition and the driver of the backward system {possesses} nonlinearity of type $f(|y|)|z|^2,$ where $f$ is any locally integrable function. We prove both the Malliavin and classical derivative of the QFBSDE and provide representations of these processes. We study a numerical approximation of this system in the sense of \cite{ImkDosReis} in which the authors assume that the drift is Lipschitz and the driver of the BSDE is quadrati…
In this paper, we discuss the solvability of backward stochastic differential equations (BSDEs) with superquadratic generators. We first prove that given a superquadratic generator, there exists a bounded terminal value, such that the associated BSDE does not admit any bounded solution. On the other hand, we prove that if the superquadratic BSDE admits a bounded solution, then there exist infinitely many bounded solutions for this BSDE. Finally, we prove the existence of a solution for Markovian BSDEs where the terminal value is a bounded continuous function of a forward stochastic differentia…
In this paper, we study reflected generalized backward doubly stochastic differential equations driven by Teugels martingales associated with Lévy process (RGBDSDELs, in short) with one continuous barrier. Under uniformly Lipschitz coefficients, we prove existence and uniqueness result by means of the penalization method and the fixed point theorem. As an application, this study allows us to give a probabilistic representation for the solutions to a class of reflected stochastic partial differential integral equations (SPDIEs, in short) with a nonlinear Neumann boundary condition.
This paper extends the results of Ma, Wu, Zhang, Zhang [11] to the context of path-dependent multidimensional forward-backward stochastic differential equations (FBSDE). By path-dependent we mean that the coefficients of the forward-backward SDE at time t can depend on the whole path of the forward process up to time t. Such a situation appears when solving path-dependent stochastic control problems by means of variational calculus. At the heart of our analysis is the construction of a decoupling random field on the path space. We first prove the existence and the uniqueness of decoupling fiel…
In this article, we are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time tau. The main motivations are giving a probabilistic representation of the Sobolev's solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when tau is the first exit time of a forward SDE from a cylindrical domain. Euler schemes and bounds for the discrete-time approximation error are provided.
Although having been developed for more than two decades, the theory of forward backward stochastic differential equations is still far from complete. In this paper, we take one step back and investigate the formulation of FBSDEs. Motivated from several considerations, both in theory and in applications, we propose to study FBSDEs in weak formulation, rather than the strong formulation in the standard literature. That is, the backward SDE is driven by the forward component, instead of by the Brownian motion noise. We establish the Feyman-Kac formula for FBSDEs in weak formulation, both in clas…
In this paper, we address the problem of existence and uniqueness of a global classical solution to a multidimensional stochastic Burgers equation without gradient-type assumptions on the force or the initial condition. The equation is first transformed to a random PDE, and then solved via the associated forward-backward SDE. Additionally, we obtain a new a priori gradient estimate valid for a large class of second-order quasilinear parabolic PDEs which becomes an important tool in our approach. Also, we study the stochastic Burgers equation in the vanishing viscosity limit.
Recent developments on financial markets have revealed the limits of Brownian motion pricing models when they are applied to actual markets. Lévy processes, that admit jumps over time, have been found more useful for applications. Thus, we suggest a Lévy model based on Forward-Backward Stochastic Differential Equations (FBSDEs) for option pricing in a Lévy-type market. We show the existence and uniqueness of a solution to FBSDEs driven by a Lévy process. This result is important from the mathematical point of view, and also, provides a much more realistic approach to option pricing.
We solve the optimal control problem of a one-dimensional reflected stochastic differential equation, whose coefficients can be path dependent. The value function of this problem is characterized by a backward stochastic partial differential equation (BSPDE) with Neumann boundary conditions. We prove the existence and uniqueness of sufficiently regular solution for this BSPDE, which is then used to construct the optimal feedback control. In fact we prove a more general result: The existence and uniqueness of strong solution for the Neumann problem for general nonlinear BSPDEs, which might be o…
Let (Y, Z) denote the solution to a forward-backward SDE. If one constructs a random walk B n from the underlying Brownian motion B by Skorohod embedding, one can show L 2 convergence of the corresponding solutions (Y n , Z n) to (Y, Z). We estimate the rate of convergence in dependence of smoothness properties, especially for a terminal condition function in C 2,$α$. The proof relies on an approximative representation of Z n and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the PDE associated to the FBSDE as well as of the finite diff…
This paper presents existence and uniqueness results for reflected backward doubly stochastic differential equations (in short RBDDSEs) in a convex domain D. Moreover, using a stochastic flow approach a probabilistic interpretation for a class of reflected SPDE's in a domain is given via such RBDSDEs. The solution is expressed as a pair (u,ν) where u is a predictable continuous process which takes values in a Sobolev space and m is a random regular measure. The bounded variation process K, component of the solution of the reflected BDSDE, controls the set when u reaches the boundary of D. This…
A method of the approximation of a coalescing Harris flow with homeomorphic stochastic flows built as solutions to SDEs w.r.t. continuous martingales with spatial parameters in the sense of Kunita is proposed. The joint convergence of forward and backward flows as diffusions is obtained, as well as the joint convergence of forward and backward transformations of the real axe under the action of the flows.
Doubly Reflected Backward SDEs Driven by $G$-Brownian Motion with Quadratic Generator
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学与交叉科学研究中心）；Frontiers Science Center for Nonlinear Expectations (Ministry of Education), Shandong University（山东大学非线性期望前沿科学中心（教育部））；Shandong Province Key Laboratory of Financial Risk, Shandong University（山东大学山东省金融风险重点实验室）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Zhongtai Securities Institute for Financial Studies, Shandong University（山东大学中泰证券金融研究院）
We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized solutions of class $C^1$. The generator $L$ is associated with a Markov process $X$ which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward pr…
In this paper, we propose a new notion of Forward--Backward Martingale Problem (FBMP), and study its relationship with the weak solution to the forward--backward stochastic differential equations (FBSDEs). The FBMP extends the idea of the well-known (forward) martingale problem of Stroock and Varadhan, but it is structured specifically to fit the nature of an FBSDE. We first prove a general sufficient condition for the existence of the solution to the FBMP. In the Markovian case with uniformly continuous coefficients, we show that the weak solution to the FBSDE (or equivalently, the solution t…
We consider a mean field game with common noise in which the diffusion coefficients may be controlled. We prove existence of a weak relaxed solution under some continuity conditions on the coefficients. We then show that, when there is no common noise, the solution of this mean field game is characterized by a McKean-Vlasov type second order backward SDE.
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 establish a connection between the strong solution to the spatially periodic Navier-Stokes equations and a solution to a system of forward-backward stochastic differential equations (FBSDEs) on the group of volume-preserving diffeomorphisms of a flat torus. We construct representations of the strong solution to the Navier-Stokes equations in terms of diffusion processes.
We obtain upper and lower Gaussian density estimates for the law of each component of the solution to a one-dimensional fully coupled forward-backward SDE (FBSDE). Our approach relies on the link between FBSDEs and quasilinear parabolic PDEs, and is fully based on the use of classical results on PDEs rather than on manipulation of FBSDEs, compared to other papers on this topic. This essentially simplifies the analysis.
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…
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.
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…
We consider a unifying framework for stochastic control problem including the following features: partial observation, path-dependence (both with respect to the state and the control), and without any non-degeneracy condition on the stochastic differential equation (SDE) for the controlled state process, driven by a Wiener process. In this context, we develop a general methodology, refereed to as the randomization method, studied in [23] for classical Markovian control under full observation, and consisting basically in replacing the control by an exogenous process independent of the driving n…
In this paper, we study the well-posedness of the Forward-Backward Stochastic Differential Equations (FBSDE) in a general non-Markovian framework. The main purpose is to find a unified scheme which combines all existing methodology in the literature, and to address some fundamental longstanding problems for non-Markovian FBSDEs. An important device is a decoupling random field that is regular (uniformly Lipschitz in its spatial variable). We show that the regulariy of such decoupling field is closely related to the bounded solution to an associated characteristic BSDE, a backward stochastic Ri…
Semilinear, $N-$dimensional stochastic differential equations (SDEs) driven by additive Lévy noise are investigated. Specifically, given $α\in\left(\frac{1}{2},1\right)$, the interest is on SDEs driven by $2α-$stable, rotation-invariant processes obtained by subordination of a Brownian motion. An original connection between the time-dependent Markov transition semigroup associated with their solutions and Kolmogorov backward equations in mild integral form is established via regularization-by-noise techniques. Such a link is the starting point for an iterative method which allows to approximat…
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…
Forward-backward stochastic differential equations (FBSDEs) have attracted significant attention since they were introduced almost 30 years ago, due to their wide range of applications, from solving non-linear PDEs to pricing American-type options. Here, we consider two new classes of multidimensional FBSDEs with distributional coefficients (elements of a Sobolev space with negative order). We introduce a suitable notion of a solution, show existence and uniqueness of a strong solution of the first FBSDE, and weak existence for the second. We establish a link with PDE theory via a nonlinear Fe…
In this paper we consider a mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn, Li, Peng and Rainer [6] to BSDEs, the existence and the uniqueness of the solution $(Y^{t,ξ}, Z^{t,ξ}, H^{t,ξ})$, $(Y^{t,x,P_ξ}, Z^{t,x,P_ξ}, H^{t,x,P_ξ})$ of the split equations are proved. The first and the second order derivatives of the process $(Y^{t,x,P_ξ}, Z^{t,x,P_ξ}, H^{t,x,P_ξ})$ with respect to $x$, the derivative of the process $(Y^{t,x,P_ξ}, Z^{t,x,P_ξ}, H^{t,x,P_ξ})$ …
We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingal…
In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify them as solutions of coupled forward-backward infinite horizon stochastic integral equations (IHSIEs), using the "substitution theorem" of stochastic differential equations with anticipating initial conditions. In general, random periodic solutions and the solutions of IHSIEs, are anticipating. For the linear noise case, with the help of the exponential dichotomy given in the multiplicative ergodic theorem, we can identify them as the solutions of infinite horizon ra…
