Stochastic Optimal Control and BSDEs with Logarithmic Growth

In this paper, we study the existence of an optimal strategy for the stochastic control of diffusion in general case and a saddle-point for zero-sum stochastic differential games. The problem is formulated as an extended BSDE with logarithmic growth in the $z$-variable and terminal value in some $L^p$ space. We also show the existence and uniqueness of solution of this BSDE.Comment: 20 page

Averaging for SDE-BSDE with null recurrent fast component Application to homogenization in a non periodic media

We establish an averaging principle for a family of solutions$(X^{\varepsilon}, Y^{\varepsilon})$ $:=$ $(X^{1,\,\varepsilon},\,X^{2,\,\varepsilon},\, Y^{\varepsilon})$ of a system of SDE-BSDEwith a null recurrent fast component $X^{1,\,\varepsilon}$. Incontrast to the classical periodic case, we can not rely on aninvariant probability and the slow forward component$X^{2,\,\varepsilon}$ cannot be approximated by a diffusion process.On the other hand, we assume that the coefficients admit a limit in a\`{C}esaro sense. In such a case, the limit coefficients may havediscontinuity. We show that we can approximate the triplet$(X^{1,\,\varepsilon},\, X^{2,\,\varepsilon},\, Y^{\varepsilon})$ bya system of SDE-BSDE $(X^1, X^2, Y)$ where $X := (X^1, X^2)$ is aMarkov diffusion which is the unique (in law) weak solution of theaveraged forward component and $Y$ is the unique solution to the averaged backward component. This is done with a backward component whosegenerator depends on the variable $z$. Asapplication, we establish an homogenization result for semilinearPDEs when the coefficients can be neither periodic nor ergodic. Weshow that the averaged BDSE is related to the averaged PDE via aprobabilistic representation of the (unique) Sobolev $W\_{d+1,\text{loc}}^{1,2}(\R\_+\times\R^d)$--solution of the limitPDEs. Our approach combines PDE methods and probabilistic argumentswhich are based on stability property and weak convergence of BSDEsin the S-topology

A class of stochastic differential equations with super-linear growth and non-Lipschitz coefficients

The purpose of this paper is to study some properties of solutions to one dimensional as well as multidimensional stochastic differential equations (SDEs in short) with super-linear growth conditions on the coefficients. Taking inspiration from \cite{BEHP, KBahlali, Bahlali}, we introduce a new {\it{local condition}} which ensures the pathwise uniqueness, as well as the non-contact property. We moreover show that the solution produces a stochastic flow of continuous maps and satisfies a large deviations principle of Freidlin-Wentzell type. Our conditions on the coefficients go beyond the existing ones in the literature. For instance, the coefficients are not assumed uniformly continuous and therefore can not satisfy the classical Osgood condition. The drift coefficient could not be locally monotone and the diffusion is neither locally Lipschitz nor uniformly elliptic. Our conditions on the coefficients are, in some sense, near the best possible. Our results are sharp and mainly based on Gronwall lemma and the localization of the time parameter in concatenated intervalsComment: in Stochastics An International Journal of Probability and Stochastic Processes, 201

Backward doubly stochastic differential equations with weak assumptions on the coefficients

In this paper, we deal with one dimensional backward doubly stochastic differential equations (BDSDEs) where the coefficient is left Lipschitz in y (may be discontinuous) and uniformly continuous in z. We obtain a generalized comparison theorem and a generalized existence theorem of BDSDEs .Comment: 17 page