On Approximation of the BSDE with Unknown Volatility in Forward Equation

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We consider the problem of the construction of the backward stochastic differential equation in the Markovian case. We suppose that the forward equation has a diffusion coefficient depending on some unknown parameter. We propose an estimator of this parameter constructed by the discrete time observations of the forward equation and then we use this estimator for approximation of the solution of the backward equation. The question of asymptotic optimality of this approximation is also discussed.