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

Optimality necessary conditions in singular stochastic control problems with nonsmooth data

AbstractThe present paper studies the stochastic maximum principle in singular optimal control, where the state is governed by a stochastic differential equation with nonsmooth coefficients, allowing both classical control and singular control. The proof of the main result is based on the approximation of the initial problem, by a sequence of control problems with smooth coefficients. We, then apply Ekeland's variational principle for this approximating sequence of control problems, in order to establish necessary conditions satisfied by a sequence of near optimal controls. Finally, we prove the convergence of the scheme, using Krylov's inequality in the nondegenerate case and the Bouleau–Hirsch flow property in the degenerate one. The adjoint process obtained is given by means of distributional derivatives of the coefficients

Lp-solutions to BSDEs with super-linear growth coefficient. Application to degenerate semilinear PDEs

We consider multidimensional backward stochastic differential equations (BSDEs). We prove the existence and uniqueness of solutions when the coefficient grow super-linearly, and moreover, can be neither locally Lipschitz in the variable y nor in the variable z. This is done with super-linear growth coefficient and a p-integrable terminal condition (p > 1). As application, we establish the existence and uniqueness of solutions to degenerate semilinear PDEs with superlinear growth generator and an Lp-terminal data, p > 1. Our result cover, for instance, the case of PDEs with logarithmic nonlinearities

On Necessary and Sufficient Conditions for Near-Optimal Singular Stochastic Controls

This paper is concerned with necessary and sufficient conditions for near-optimal singular stochastic controls for systems driven by a nonlinear stochastic differential equations (SDEs in short). The proof of our result is based on Ekeland's variational principle and some delicate estimates of the state and adjoint processes. This result is a generalization of Zhou's stochastic maximum principle for near-optimality to singular control problem.Comment: 19 pages, submitted to journa

Adaptive importance sampling with forward-backward stochastic differential equations

We describe an adaptive importance sampling algorithm for rare events that is based on a dual stochastic control formulation of a path sampling problem. Specifically, we focus on path functionals that have the form of cumulate generating functions, which appear relevant in the context of, e.g.~molecular dynamics, and we discuss the construction of an optimal (i.e. minimum variance) change of measure by solving a stochastic control problem. We show that the associated semi-linear dynamic programming equations admit an equivalent formulation as a system of uncoupled forward-backward stochastic differential equations that can be solved efficiently by a least squares Monte Carlo algorithm. We illustrate the approach with a suitable numerical example and discuss the extension of the algorithm to high-dimensional systems

BSDE associated with Lévy processes and application to PDIE

We deal with backward stochastic differential equations (BSDE for short) driven by Teugel's martingales and an independent Brownian motion. We study the existence, uniqueness and comparison of solutions for these equations under a Lipschitz as well as a locally Lipschitz conditions on the coefficient. In the locally Lipschitz case, we prove that if the Lipschitz constant LN behaves as log(N) in the ball B(0,N), then the corresponding BSDE has a unique solution which depends continuously on the on the coefficient and the terminal data. This is done with an unbounded terminal data. As application, we give a probabilistic interpretation for a large class of partial differential integral equations (PDIE for short)