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In this paper, we consider the problem of numerical solution of the system of forward– backward stochastic differential equations and its Cauchy problem for a quasilinear parabolic equation. We propose a layer method of solving the Cauchy problem with simultaneous estimation of the solution gradient, which is based on the probabilistic representation of the solution. We consider the ways of constructing layer methods for nonlinear boundary value problems. The system of stochastic differential equations is solved by the Euler method using the preliminarily obtained numerical solution of a quasilinear equation and its gradient.

Numerical solution of quasilinear parabolic equations and backward stochastic differential equations