GENERAL COUPLED MEAN-FIELD REFLECTED FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper we consider general coupled mean-field reflected forward-backward stochastic differential equations(FBSDEs),whose coefficients not only depend on the solution but also on the law of the solution.The first part of the paper is devoted to the existence and the uniqueness of solutions for such general mean-field reflected backward stochastic differential equations(BSDEs)under Lipschitz conditions,and for the one-dimensional case a comparison theorem is studied.With the help of this comparison result,we prove the existence of the solution for our mean-field reflected forward-backward stochastic differential equation under continuity assumptions.It should be mentioned that,under appropriate assumptions,we prove the uniqueness of this solution as well as that of a comparison theorem for mean-field reflected FBSDEs in a non-trivial manner.

EXPONENTIAL STABILITY FOR NONLINEAR HYBRID STOCHASTIC PANTOGRAPH EQUATIONS AND NUMERICAL APPROXIMATION

The paper develops exponential stability of the analytic solution and convergence in probability of the numerical method for highly nonlinear hybrid stochastic pantograph equation. The classical linear growth condition is replaced by polynomial growth conditions, under which there exists a unique global solution and the solution is almost surely exponentially stable. On the basis of a series of lemmas, the paper establishes a new criterion on convergence in probability of the Euler-Maruyama approximate solution. The criterion is very general so that many highly nonlinear stochastic pantograph equations can obey these conditions. A highly nonlinear example is provided to illustrate the main theory.

FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH STOPPING TIME

The existence and uniqueness results of fully coupled forward-backward stochastic differential equations with stopping time (unbounded) is obtained. One kind of comparison theorem for this kind of equations is also proved.

Solutions to general forward-backward doubly stochastic differential equations

A general type of forward-backward doubly stochastic differential equations （FBDSDEs） is studied. It extends many important equations that have been well studied, including stochastic Hamiltonian systems. Under some much weaker monotonicity assumptions, the existence and uniqueness of measurable solutions are established with a incthod of continuation. Furthermore, the continuity and differentiability of the solutions to FBDSDEs depending on parameters is discussed.

The Semi-implicit Euler Method for Stochastic Pantograph Equations with Jumps

In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.

A Mean-Field Stochastic Maximum Principle for Optimal Control of Forward-Backward Stochastic Differential Equations with Jumps via Malliavin Calculus

This paper considers a mean-field type stochastic control problem where the dynamics is governed by a forward and backward stochastic differential equation (SDE) driven by Lévy processes and the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly non-Markovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed.

Issues in the Influence of Ito-type Noise on the Oscillation of Solutions of Delay Differential Pantograph Equations

In this paper, a deterministic delay differential pantograph equation （DDPE） with an unbounded memory is stochastically perturbed by an Ito-type noise. The contribution of white noise to the oscillatory behaviour of the new stochastic delay differential pantograph equation （SDDPE） is investigated. It is established that under certain conditions and with a highly positive probability, the new stochastic delay differential pantograph equation has an oscillatory solution influenced by the presence of the noise. This is not possible with the original deterministic system which has a non-oscillatory solution due to the absence of noise.

L^(p)-Estimate for Linear Forward-Backward Stochastic Differential Equations

This paper is concerned with coupled linear forward-backward stochastic differential equations(FBSDEs,for short).When the homogeneous coefficients are deterministic(the non-homogeneous coefficients can be random),we obtain an L^(P)-result(p>2),including the existence and uniqueness of the p-th power integrable solution,a p-th power estimate,and a related continuous dependence property of the solution on the coefficients,for coupled linear FBSDEs in the monotonicity framework over large time intervals.In order to get rid of the stubborn constraint commonly existing in the literature,i.e.,the Lipschitz constant of σ with respect to z is very small,we introduce a linear transformation to overcome the difficulty on small intervals,and then"splice"the L^(P)-results obtained on many small intervals to yield the desired one on a large interval.

Existence of Solutions for Forward-Backward Stochastic Differential Equations with Jumps and Non-Lipschitzian Coefficients

This paper studies for ward-back ward differential equations with Poisson jumps and with stopping time as termination. Under some weak monotonicity conditions and for non-Lipschitzian coefficients, the existence and uniqueness of solutions are proved via a purely probabilistic approach, while a priori estimate is given. Here, we allow the forward equation to be degenerate.

STRONG PREDICTOR-CORRECTOR METHODS FOR STOCHASTIC PANTOGRAPH EQUATIONS

The paper introduces a new class of numerical schemes for the approximate solutions of stochastic pantograph equations. As an effective technique to implement implicit stochastic methods, strong predictor-corrector methods （PCMs） are designed to handle scenario simulation of solutions of stochastic pantograph equations. It is proved that the PCMs are strong convergent with order 1/2.Linear M^-stabiiity of stochastic pantograph equationsand the PCMs are researched in the paper. Sufficient conditions of MS-unstability of stochastic pantograph equations and MS-stability of the PCMs are obtained, respectively. Numerical experiments demonstrate these theoretical results.

Forward-backward doubly stochastic differential equations and related stochastic partial differential equations

The notion of bridge is introduced for systems of coupled forward-backward doubly stochastic differential equations （FBDSDEs）. It is proved that if two FBDSDEs are linked by a bridge, then they have the same unique solvability. Consequently, by constructing appropriate bridges, we obtain several classes of uniquely solvable FBDSDEs. Finally, the probabilistie interpretation for the solutions to a class of quasilinear stochastic partial differential equations （SPDEs） combined with algebra equations is given. One distinctive character of this result is that the forward component of the FBDSDEs is coupled with the backvzard variable.

A Type of General Forward-Backward Stochastic Differential Equations and Applications

The authors discuss one type of general forward-backward stochastic differential equations (FBSDEs) with Ito's stochastic delayed equations as the forward equations and anticipated backward stochastic differential equations as the backward equations.The existence and uniqueness results of the general FBSDEs are obtained.In the framework of the general FBSDEs in this paper,the explicit form of the optimal control for linear-quadratic stochastic optimal control problem with delay and the Nash equilibrium point for nonzero sum differential games problem with delay are obtained.

Well-Posedness of Fully Coupled Linear Forward-Backward Stochastic Differential Equations

This paper studies the well-posedness of fully coupled linear forward-backward stochastic differential equations (FBSDEs). The authors introduce two main methods-the method of continuation under monotonicity conditions and the unified approach-to ensure the existence and uniqueness of solutions of fully coupled linear FBSDEs. The authors show that the first method (the method of continuation under monotonicity conditions) can be deduced as a special case of the second method (the unified approach). An example is given to illustrate it in linear FBSDEs case. And then, a linear transformation method in virtue of the non-degeneracy of transformation matrix is introduced for cases that the linear FBSDEs can not be dealt with by the the method of continuation under monotonicity conditions and the unified approach directly. As a powerful supplement to the the method of continuation under monotonicity conditions and the unified approach, linear transformation method overall develops the well-posedness theory of fully coupled linear forward-backward stochastic differential equations which have potential applications in optimal control and partial differential equation theory.

Fully Coupled Forward-Backward Stochastic Functional Differential Equations and Applications to Quadratic Optimal Control

This paper considers 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. The authors 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.

Euler-type schemes for weakly coupled forward-backward stochastic differential equations and optimal convergence analysis

We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations （FBSDEs）. Although the schemes share some common features with the ones proposed by C. Bender and J. Zhang [Ann. Appl. Probab., 2008, 18： 143-177], less computational work is needed for our method. For both our schemes and the ones proposed by Bender and Zhang, we rigorously obtain first-order error estimates, which improve the half-order error estimates of Bender and Zhang. Moreover, numerical tests are given to demonstrate the first-order accuracy of the schemes.

A Numerical Method and its Error Estimates for the Decoupled Forward-Backward Stochastic Differential Equations

In this paper,a new numerical method for solving the decoupled forwardbackward stochastic differential equations(FBSDEs)is proposed based on some specially derived reference equations.We rigorously analyze errors of the proposed method under general situations.Then we present error estimates for each of the specific cases when some classical numerical schemes for solving the forward SDE are taken in the method;in particular,we prove that the proposed method is second-order accurate if used together with the order-2.0 weak Taylor scheme for the SDE.Some examples are also given to numerically demonstrate the accuracy of the proposed method and verify the theoretical results.

Linear-quadratic generalized Stackelberg games with jump-diffusion processes and related forward-backward stochastic differential equations

A kind of linear-quadratic Stackelberg games with the multilevel hierarchy driven by both Brownian motion and Poisson processes is considered.The Stackelberg equilibrium is presented by linear forward-backward stochastic differential equations(FBSDEs)with Poisson processes(FBSDEPs)in a closed form.By the continuity method,the unique solvability of FBSDEPs with a multilevel self-similar domination-monotonicity structure is obtained.

Infinite Horizon Forward-Backward Doubly Stochastic Differential Equations and Related SPDEs

A type of infinite horizon forward-backward doubly stochastic differential equations is studied.Under some monotonicity assumptions,the existence and uniqueness results for measurable solutions are established by means of homotopy method.A probabilistic interpretation for solutions to a class of stochastic partial differential equations combined with algebra equations is given.A significant feature of this result is that the forward component of the FBDSDEs is coupled with the backward variable.

A FIRST-ORDER NUMERICAL SCHEME FOR FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN BOUNDED DOMAINS

We propose a novel numerical scheme for decoupled forward-backward stochastic differ- ential equations （FBSDEs） in bounded domains, which corresponds to a class of nonlinear parabolic partial differential equations with Dirichlet boundary conditions. The key idea is to exploit the regularity of the solution （Yt,Zt） with respect to Xt to avoid direct ap- proximation of the involved random exit time. Especially, in the one-dimensional case, we prove that the probability of Xt exiting the domain within At is on the order of O（（△t）ε exp（--1/（△t）2ε））, if the distance between the start point X0 and the boundary is 1 g at least on the order of O（△t）^1/2-ε ） for any fixed c 〉 0. Hence, in spatial discretization, we set the mesh size △x - （9（（At）^1/2-ε ）, so that all the interior grid points are sufficiently far from the boundary, which makes the error caused by the exit time decay sub-exponentially with respect to △t. The accuracy of the approximate solution near the boundary can be guaranteed by means of high-order piecewise polynomial interpolation. Our method is developed using the implicit Euler scheme and cubic polynomial interpolation, which leads to an overall first-order convergence rate with respect to △t.

非线性随机Pantograph微分方程及其θ-方法的均方渐近稳定性

本文主要研究了非线性随机Pantograph微分方程,讨论了其零解的均方渐近稳定性并给出了零解均方渐近稳定的充分条件.在本文的第三部分,我们将随机θ-方法应用于这类问题,获得了数值解均方渐近稳定条件.