ON SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS,WITH UNBOUNDED STOPPING TIMES AS TERMINAL AND WITH NON-LIPSCHITZ COEFFICIENTS,AND PROBABILISTIC INTERPRETATION OF QUASI-LINEAR ELLIPTIC TYPE INTEGRO-DIFFERENTIAL EQUATIO

The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi_linear elliptic type integro_differential equations is obtained.

Relationship Between General MP and DPP for the Stochastic Recursive Optimal Control Problem with Jumps

This paper is concerned with the relationship between general maximum principle and dynamic programming principle for the stochastic recursive optimal control problem with jumps,where the control domain is not necessarily convex.Relations among the adjoint processes,the generalized Hamiltonian function and the value function are proven,under the assumption of a smooth value function and within the framework of viscosity solutions,respectively.Some examples are given to illustrate the theoretical results.

Convex Concentration for Some Additive Functionals of Jump Stochastic Differential Equations

Using forward-backward stochastic calculus, we prove convex concentration inequalities for some additive functionals of the solution of stochastic differential equations with jumps admitting an invariant probability measure. As a consequence, transportation-information inequalities are obtained and bounds on option prices for interest rate derivatives are given as an application.

An Accurate Numerical Scheme for Mean-Field Forward and Backward SDEs with Jumps

In this work,we propose an explicit second order scheme for decoupled mean-field forward backward stochastic differential equations with jumps.The sta-bility and the rigorous error estimates are presented,which show that the proposed scheme yields a second order rate of convergence,when the forward mean-field stochastic differential equation is solved by the weak order 2.0 Itˆo-Taylor scheme.Numerical experiments are carried out to verify the theoretical results.

Mean-Field Maximum Principle for Optimal Control of Forward–Backward Stochastic Systems with Jumps and its Application to Mean-Variance Portfolio Problem

We study mean-field type optimal stochastic control problem for systems governed by mean-field controlled forward-backward stochastic differential equations with jump processes,in which the coefficients depend on the marginal law of the state process through its expected value.The control variable is allowed to enter both diffusion and jump coefficients.Moreover,the cost functional is also of mean-field type.Necessary conditions for optimal control for these systems in the form of maximum principle are established by means of convex perturbation techniques.As an application,time-inconsistent mean-variance portfolio selectionmixed with a recursive utility functional optimization problem is discussed to illustrate the theoretical results.

A New Representation for Second Order Stochastic Integral-differential Operators and Its Applications

In this paper, we＇ll prove new representation theorems for a kind of second order stochastic integral- differential operator by stochastic differential equations （SDEs） and backward stochastic differential equations （BSDEs） with jumps, and give some applications.

A NEW SECOND ORDER NUMERICAL SCHEME FOR SOLVING DECOUPLED MEAN-FIELD FBSDES WITH JUMPS

In this paper, we consider the numerical solution of decoupled mean-field forward backward stochastic differential equations with jumps (MFBSDEJs). By using finite difference approximations and the Gaussian quadrature rule, and the weak order 2.0 Itô-Taylor scheme to solve the forward mean-field SDEs with jumps, we propose a new second order scheme for MFBSDEJs. The proposed scheme allows an easy implementation. Some numerical experiments are carried out to demonstrate the stability, the effectiveness and the second order accuracy of the scheme.