Etienne Pardoux

Adapted solution of a backward stochastic differential equation

Abstract Let Wt; t ϵ [0, 1] be a standard k-dimensional Weiner process defined on a probability space ( Ω, F, P ), and let Ft denote its natural filtration. Given a F1 measurable d-dimensional random vector X, we look for an adapted pair of processes {x(t), y(t); t ϵ [0, 1]} with values in Rd and Rd×k respectively, which solves an equation of the form: x(t) + ∫ t 1 f(s, x(s), y(s)) ds + ∫ t 1 [g(s, x(s)) + y(s)] dW s = X. A linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process. We also generalize our results to the following equation: x(t) + ∫ t 1 f(s, x(s), y(s)) ds + ∫ t 1 g(s, x(s)) + y(s)) dW s = X under rather restrictive assumptions on g.

Stochastic Calculus with Anticipating Integrands

SummaryWe study the stochastic integral defined by Skorohod in [24] of a possibly anticipating integrand, as a function of its upper limit, and establish an extended Itô formula. We also introduce an extension of Stratonovichs integral, and establish the associated chain rule. In all the results, the adaptedness of the integrand is replaced by a certain smoothness requirement.

Backward stochastic differential equations and integral-partial differential equations

We consider a backward stochastic differential equation, whose data (the final condition and the coefficient) are given functions of a jump-diffusion process. We prove that under mild conditions the solution of the BSDE provides a viscosity solution of a system of parabolic integral-partial differential equations. Under an additional assumption, that system of equations is proved to have a unique solution, in a given class of continuous functions

Lp solutions of backward stochastic differential equations

In this paper, we are interested in solving backward stochastic differential equations (BSDEs for short) under weak assumptions on the data. The first part of the paper is devoted to the development of some new technical aspects of stochastic calculus related to BSDEs. Then we derive a priori estimates and prove existence and uniqueness of solutions in Lp p>1, extending the results of El Karoui et al. (Math. Finance 7(1) (1997) 1) to the case where the monotonicity conditions of Pardoux (Nonlinear Analysis; Differential Equations and Control (Montreal, QC, 1998), Kluwer Academic Publishers, Dordrecht, pp. 503-549) are satisfied. We consider both a fixed and a random time interval. In the last section, we obtain, under an additional assumption, an existence and uniqueness result for BSDEs on a fixed time interval, when the data are only in L1.

BSDEs, weak convergence and homogenization of semilinear PDEs

In these lectures, we present the theory of backward stochastic differential equations, and its connection with solutions of semilinear second order partial differential equations of parabolic and elliptic type. This connection provides a probabilistic tool for studying solutions of semilinear PDEs. We apply our results to the proof of the homogenization result for such PDEs, both with periodic and random coefficients. For that purpose, we need to present the theory of weak limits of solutions of backward stochastic differential equations. We also present a complete probabilistic proof, under apparently minimal assumptions, of the homogenization result of linear second order PDEs.

Backward doubly stochastic differential equations and systems of quasilinear SPDEs

SummaryWe introduce a new class of backward stochastic differential equations, which allows us to produce a probabilistic representation of certain quasilinear stochastic partial differential equations, thus extending the Feynman-Kac formula for linear SPDEs.

Discretization and simulation of stochastic differential equations

We discuss both pathwise and mean-square convergence of several approximation schemes to stochastic differential equations. We then estimate the corresponding speeds of convergence, the error being either the mean square error or the error induced by the approximation on the value of the expectation of a functional of the solution. We finally give and comment on a few comparative simulation results.

On Poisson equation and diffusion approximation 2

A Poisson equation in d for the elliptic operator corresponding to an ergodic diffusion process is considered. Existence and uniqueness of its solution in Sobolev classes of functions is established along with the bounds for its growth. This result is used to study a diffusion approximation for two-scaled diffusion processes using the method of corrector; the solution of a Poisson equation serves as a corrector.

Optimal Control for Partially Observed Diffusions

Stochastic control problems are considered in which a state process

Équations du filtrage non linéaire de la prédiction et du lissage

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