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- Etienne Pardoux
- 2010

We introduce a new class of backward stochastic differential equations , which allows us to produce a probabilistic representation of certain quasi-linear stochastic partial differential equations, thus extending the Feynman-Kac formula for linear SPDE's.

- Etienne Pardoux
- 1996

Introduction The aim of this set of lectures is to present the theory of backward stochastic differential equations, in short BSDEs, and its connections with viscosity solutions of systems of semi– linear second order partial differential equations of parabolic and elliptic type, in short PDEs. Linear BSDEs have appeared long time ago, both as the equations… (More)

This paper studies, under some natural monotonicity conditions, the theory (existence and uniqueness, a priori estimate, continuous dependence on a parameter) of forward–backward stochastic differential equations and their connection with quasilinear parabolic partial differential equations. We use a purely probabilistic approach, and allow the forward… (More)

Let L be a second-order partial differential operator in R e. Let R e be the finite union of disjoint polyhedra. Suppose that the diffusion matrix is everywhere non singular and constant on each polyhedron, and that the drift coefficient is bounded and measurable. We show that the martingale problem associated with L is well-posed.

- E. PARDOUX
- 2005

We study the Poisson equation Lu + f = 0 in R d , where L is the infinitesimal generator of a diffusion process. In this paper, we allow the second-order part of the generator L to be degenerate, provided a local condition of Doeblin type is satisfied, so that, if we also assume a condition on the drift which implies recurrence, the diffusion process is… (More)

We study a new class of backward stochastic dierential equations, which involves the integral with respect to a continuous increasing process. This allows us to give a probabilistic formula for solutions of semilinear partial dierential equations with Neumann boundary condition, where the boundary condition itself is nonlinear. We consider both parabolic… (More)

- C Donati-Martin, E Pardoux
- 1991

We study reflected solutions of a nonlinear heat equation on the spatial interval [0, 1] with Dirichlet boundary conditions, driven by space-time white noise. The nonlinearity appears both in the drift and in the diffusion coefficient. Roughly speaking, at any point (t, x) where the solution u(t, x) is strictly positive it obeys the equation, and at a point… (More)

- E. PARDOUX
- 1986

It is shown that if a diffusion process, {Xt: 0 < t < 1}, on Rd satisfies dXt = b(t, Xt) dt + a(t, Xt) dwt then the reversed process, {Xt: 0 < t < 1} where Xt = Xl t , is again a diffusion with drift b and diffusion coefficient a, provided some mild conditions on b, a, and p(, the density of the law of X(, hold. Moreover b and a are identified. 1.… (More)

It is well-known under the name of 'periodic homogenization' that, under a centering condition of the drift, a periodic diffusion process on R d converges, under diffusive rescaling, to a d-dimensional Brownian motion. Existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion. In this paper, we… (More)