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Numerical methods for forward-backward stochastic differential equations. (English) Zbl 0861.65131
This paper presents numerical methods for approximating the solutions of forward-backward stochastic differential equations of the form \[ X_t=x+\int^t_0 b(s,X_s,Y_s,Z_s)ds+ \int^t_0\sigma(s,X_s,Y_s)dW_s, \] \[ Y_t=g(X_T)+ \int^T_t\widehat b(s,X_s,Y_s,Z_s)ds+ \int^T_t\widehat\sigma(s,X_s,Y_s,Z_s)dW_s, \] where \(t\in[0,T]\), \(\{W_t\}\) is a \(d\)-dimensional Brownian motion, \((X,Y,Z)\) takes values in \(\mathbb{R}^m\times\mathbb{R}^m\times\mathbb{R}^{m\times d}\), and \(b\), \(\widehat b\), \(\sigma\), \(\widehat\sigma\), and \(g\) are smooth. The bulk of the paper is devoted to introducing approximations needed to devise the numerical methods, proving that the resulting approximate solutions converge to the actual solution, and establishing the rate of this convergence.

MSC:
65C99 Probabilistic methods, stochastic differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
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