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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
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