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Numerical method for backward stochastic differential equations. (English) Zbl 1017.60074
Let \(W\) be a \(d\)-dimensional Brownian motion. The authors develop a new method of approximating solutions \(Y\) of the multidimensional backward stochastic differential equation (BSDE) \[ dY_t= -f(t, Y_t)dt+ Z_t dW_t,\quad t\in [0,T], \] with a continuous driver \(f\) which is Lipschtz in the \(y\)-variable and independent of \(z\). As a consequence the interest in numerical treatments of such equations in finance, several numerical approximating methods for BSDEs have already been developed, cf. J. Douglas jun., J. Ma and P. Protter [Ann. Appl. Probab. 6, No. 3, 940-968 (1996; Zbl 0861.65131)], V. Bally [in: Backward stochastic differential equations. Pitman Res. Notes Math. Ser. 364, 177-191 (1997; Zbl 0889.60068)], D. Chevance [in: Numerical methods in finance, 232-244 (1997; Zbl 0898.90031)] and V. Bally, G. Pagès and J. Printems [Monte Carlo Methods Appl. 7, No. 1/2, 21-33 (2001)]. Another type of approximating BSDEs uses the discretization of the filtration [e.g. F. Coquet, V. Mackevičius and J. Mémin, Stochastic Processes Appl. 75, No. 2, 235-248 (1998; Zbl 0932.60047)].
In their present paper the authors propose a discretization of the above BSDE where the Brownian motion is replaced by a simple random walk. Although the method of the authors uses ideas similar to those used in the above papers, in that they also approximate the Brownian motion by a discrete process, their main feature is that they do not need the convergence of the discretized filtrations to the original Brownian one in order to get the weak convergence in the Skorokhod topology of the solutions. Such a relaxation reduces the complexity in the construction of approximating solutions.

60H20 Stochastic integral equations
65C30 Numerical solutions to stochastic differential and integral equations
Full Text: DOI
[1] ANTONELLI, F. and KOHATSU-HIDA, A. (2000). Filtration stability of backward SDE’s. Stochastic Anal. Appl. 18 11-37. · Zbl 0953.60044
[2] BALLY, V. (1997). Approximation scheme for solutions of BSDE. Backward Stochastic Differential Equations (N. El Karoui and L. Mazliak, eds.) 177-191. Pitman, London. · Zbl 0889.60068
[3] BALLY, V., PAGÈS, G. and PRINTEMS, J. (2001). A stochastic quantization method for nonlinear problems. Monte Carlo Methods Appl. 7 21-34. · Zbl 1035.65008
[4] BUCKDAHN, R. (2001). Backward stochastic differential equations driven by a martingale. Preprint 93-05, Univ. Provence.
[5] CHEVANCE, D. (1997). Discrétisation des équations différentieles stochastiques rétrogrades. In Numerical Methods in Finance (L. C. G. Rogers and D. Talay, eds.) 232-244. Cambridge Univ. Press. · Zbl 0898.90031
[6] COQUET, F., MACKEVI CIUS, V. and MÉMIN, J. (1998). Stability in D of martingales and backward equations under discretization of filtration. Stochastic Process. Appl. 75 235- 248. · Zbl 0932.60047
[7] DOUGLAS, J., MA, J. and PROTTER, P. (1996). Numerical methods for forward-backward stochastic differential equations. Ann. Appl. Probab. 6 940-968. · Zbl 0861.65131
[8] EL KAROUI, N., PENG, S. and QUENEZ, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1-71. · Zbl 0884.90035
[9] EL KAROUI, N. and QUENEZ, M. C. (1997). Nonlinear pricing theory and backward stochastic differential equations. Financial Mathematics. Lecture Notes in Math. 1656 191-246. Springer, New York. · Zbl 0904.90010
[10] FREIDLIN, M. (1985). Functional Integration and Partial Differential Equations. Princeton Univ. Press. · Zbl 0568.60057
[11] JACOD, J., MÉMIN, J. and METIVIER, M. (1983). On tightness and stopping times. Stochastic Process. Appl. 14 109-146. · Zbl 0501.60029
[12] LEPELTIER, J. P. and SAN MARTIN, J. (1997). Backward stochastic differential equations with continuous generator. Statist. Probab. Lett. 32 425-430. · Zbl 0904.60042
[13] MA, J., PROTTER, P. and YONG, J. (1994). Solving forward-backward stochastic differential equations explicitly-a four step scheme. Probab. Theory Relat/Fields 98 339-359. · Zbl 0794.60056
[14] PARDOUX, E. and PENG, S. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55-61. · Zbl 0692.93064
[15] PARDOUX, E. and PENG, S. (1992). Backward stochastic differential equation and quasilinear parabolic partial differential equations. Stochastic Partial Differential Equations and Their Applications. Lecture Notes in Control and Inform. Sci. 176 200-217. Springer, New York. · Zbl 0766.60079
[16] PROTTER, P. (1990). Stochastic Integration and Differential Equations. Springer, New York. · Zbl 0694.60047
[17] WEST LAFAYETTE, INDIANA 47907-1395 P. PROTTER ORIE CORNELL UNIVERSITY ITHACA, NEW YORK 14853-3801 E-MAIL: protter@orie.cornell.edu
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