Volume 20, Issue 3

Abstract. A large deviations principle for the backward stochastic differential equations connected with solutions of Itô equations with small diffusion and coefficients depending on a small parameter is proved. It is not required the existence of the limits of coefficients by tending small parameter to zero. The functions can have oscillation under the first variable. The uniform convergence on compacts of the solutions of the quasilinear parabolic equations of the second order with a small parameter by the higher derivative to the generalized solution of nonlinear equation with the partial derivatives of the first order are proved for the justification of this principle of large deviations.

Abstract. We construct absolute continuous stochastic processes that converge to anisotropic fractional and multifractional Brownian sheets in Besov-type spaces.

Abstract. Let X 1 and X 2 be two Hilbert-valued Itô processes with respect to Lévy noise. Let P 1 and P 2 be the probability measures induced by X 1 and X 2 on the Skorohod space. In the paper a formula is derived for the Kakutani–Hellinger affinity of P 1 and P 2 . Since Lévy processes of pure jump type can be described by Poisson random measures, the results holds also for Itô processes driven by Lévy processes of pure jump type.

Abstract. We study relaxed stochastic control problems where the state equation is a forward backward doubly stochastic differential equation with Poisson jumps, where the set of strict (classical) controls need not be convex and the diffusion coefficient and the generator coefficient depends on the terms control. In this paper, we introduce a new approach to solve this open problem, the main result is necessary conditions as well as a sufficient for optimality in the form of a relaxed maximum principle, with application to linear quadratic stochastic control problem with random jumps.

Abstract. In this paper, we prove two main results. The first one is to prove the regularity of fractional derivatives of local time of symmetric stable process with index ; our result is similar to that of Marcus and Rosen (1992) for local time. The second result is to give a -variation of fractional derivatives of local time of symmetric stable process with index . Our approach is similar to that of Eisenbaum (2000) for local time.