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Approximation scheme for solutions of BSDE. (English) Zbl 0889.60068
El Karoui, Nicole (ed.) et al., Backward stochastic differential equations. Harlow: Longman. Pitman Res. Notes Math. Ser. 364, 177-191 (1997).
It is considered an approximation scheme for the $$r$$-dimensional system of backward stochastic differential equations $Y_t+ \int^T_t Z_s dB_s= \xi+ \int^T_t f(s,\omega, Y_s, Z_s)ds,$ where $$B$$ is a $$d$$-dimensional Brownian motion, $$\xi$$ is a square integrable, $$F_T$$ measurable random variable and $$f$$ is Lipschitz continuous and linear growth in $$y$$, $$z$$ and continuous adapted in $$t$$, $$\omega$$. The equation has a unique solution as proved by E. Pardoux and S. Peng [in: Stochastic partial differential equations and their applications. Lect. Notes Control Inf. Sci. 176, 200-217 (1992; Zbl 0766.60079)]. An approximation scheme is given. The main idea is to use in this approximation a randomized time net instead of the discrete one, which seems to be natural but creates serious difficulties. This allows to write an equation which is satisfied by approximation $\overline Y_t+ \int^T_t\overline Z_s dB_s= \xi+ \int^T_t f(s,\omega,\overline Y_s,\overline Z_s)ds+ \int^T_t{1\over \lambda} f(s,\omega,\overline Y_s,\overline Z_s)d\widehat N(T- s),$ where $$N$$ is a Poisson process with parameter $$\lambda$$ (a random net is a sequence of jumps of this process) and $$\widehat N(t)= N(t)- \lambda t$$. Hence, the error is given by the integral with respect to $$\widehat N$$, so it is easy to estimate the error.
For the entire collection see [Zbl 0866.00052].

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 65C99 Probabilistic methods, stochastic differential equations