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Averaging of backward stochastic differential equations, with application to semilinear PDE’s. (English) Zbl 0891.60053

The authors prove an averaging result for a backward stochastic differential equation of the form \[ Y^\varepsilon_t= g(X_T^{1, \varepsilon}) +\int^T_t f(X_s^{1, \varepsilon}, X_s^{2, \varepsilon}, Y_s^\varepsilon)\, ds-\int^T_t Z_s^\varepsilon\, dW_s, \] where \((X^{1, \varepsilon}, X^{2,\varepsilon})\) is a diffusion process with an infinitesimal generator of the form \(\varepsilon^{-2} L_2+ [\varepsilon^{-1} F+G] \nabla\). The limit process solves the backward stochastic differential equation \[ Y_t= g(X^1_T) +\int^T_t \overline f(X^1_s,Y_s)\, ds-\int^T_t Z_s\, dB_s, \] where \(\overline f(x_1,y) =\int f(x_1,x_2,y) \mu(dx_2)\), and \(\mu\) is the unique invariant measure of \(X^{2, \varepsilon}\). As an application they obtain an averaging result for a system of semilinear parabolic partial differential equations. The weak convergence of the solution of the BSDE is proved using the topology of Meyer and Zheng. An extended uniqueness result for BSDE’s is applied to identify the limit.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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