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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].

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93E03 Stochastic systems in control theory (general)
35R60 PDEs with randomness, stochastic partial differential equations