Anticipated Generalized Backward Doubly Stochastic Differential Equations

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process but also include generators depending on the anticipated terms of the solution (Y, Z). Firstly, we prove the existence and uniqueness theorem for AGBDSDEs. Further, two comparison theorems are obtained after finding a new comparison theorem for GBDSDEs.

Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations

<abstract><p>We consider several aspects of conjugating symmetry methods, including the method of invariants, with an asymptotic approach. In particular we consider how to extend to the stochastic setting several ideas which are well established in the deterministic one, such as conditional, partial and asymptotic symmetries. A number of explicit examples are presented.</p></abstract>

STUDY OF MODELS WITHOUT JUMPS WITH RESPECT TO FRACTIONAL BROWNIAN MOTION

In this paper, we study some models without jumps of stochastic differential equations directed by a fractional Brownian motion.