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 probabilistic 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 backward variable.
＜正＞The partially observed control problem is considered for backward doubly stochastic systems with control en...
ZHU Qingfeng~(1,2),WANG Tianxiao~2,SHI Yufeng~2 1.School of Statistics and Mathematics,Shandong University of Finance,Jinan 250014,P.R.China 2.School of Mathematics,Shandong University,Jinan 250100,P.R.China
In this work the existence of solutions of one-dimensional backward doubly stochastic differential equations(BDSDEs) with coefficients left-Lipschitz in y(may be discontinuous) and Lipschitz in z is studied.Also,the associated comparison theorem is obtained.
Backward doubly stochastic differential equations driven by Brownian motions and Poisson process(BDSDEP) with non-Lipschitz coeffcients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations(SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.
This paper studies linear quadratic games problem for stochastic Volterra integral equations(SVIEs in short) where necessary and sufficient conditions for the existence of saddle points are derived in two different ways.As a consequence,the open problems raised by Chen and Yong(2007) are solved.To characterize the saddle points more clearly,coupled forward-backward stochastic Volterra integral equations and stochastic Fredholm-Volterra integral equations are introduced.Compared with deterministic game problems,some new terms arising from the procedure of deriving the later equations reflect well the essential nature of stochastic systems.Moreover,our representations and arguments are even new in the classical SDEs case.
This paper is devoted to the unique solvability of backward stochastic Volterra integral equations (BSVIEs, for short), in terms of both M-solution and the adapted solutions. We prove the existence and uniqueness of M-solutions of BSVIEs in Lp (1 < p < 2), which extends the existing results on M-solutions. The unique solvability of adapted solutions of BSVIEs in Lp (p > 1) is also considered, which also generalizes the results in the existing literature.
