BSDEs, weak convergence and homogenization of semilinear PDEs.

*(English)*Zbl 0959.60049
Clarke, F. H. (ed.) et al., Nonlinear analysis, differential equations and control. Proceedings of the NATO Advanced Study Institute and séminaire de mathématiques supérieures, Montréal, Canada, July 27-August 7, 1998. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 528, 503-549 (1999).

Backward stochastic differential equations (BSDEs) were introduced by the author and S. G. Peng [Syst. Control Lett. 44, No. 1, 55-61 (1990; Zbl 0692.93064)]. In these notes the author presents the basic facts on these equations and its connections with partial differential equations (PDEs) and homogenization problems. Section 2 contains the main result on existence and uniqueness of solutions to BSDEs under Lipschitz type conditions on the nonlinear coefficient. A comparison theorem is also proved. Section 3 deals with the relation between BSDEs and viscosity solutions of systems of semilinear second-order partial differential equations of parabolic type. In Section 4 one studies BSDEs with random terminal time, and the results of this section are applied to provide a probabilistic interpretation to systems of semilinear elliptic PDEs in an open bounded subset of \({\mathbf R}^d\) with Dirichlet boundary conditions. Section 6 deals with the weak convergence of the solutions \(Y^n\) of a sequence of BSDEs assuming the weak convergence of the sequence of diffusion processes \(X^n\) which are coupled with the BSDEs. Finally, in Section 7 the weak convergence of a sequence of diffusion processes \(X^n\) is established in the case of homogenization of diffusions with periodic coefficients. The results of Sections 6 and 7 lead to homogenization results for semilinear systems of PDEs with periodic coefficients.

For the entire collection see [Zbl 0913.00037].

For the entire collection see [Zbl 0913.00037].

Reviewer: David Nualart (Barcelona)