Numerical Analysis of Stochastic Differential Systems and its Applications in Finance

Abstract

In this note, we provide a survey of recentresults on numerical analysis of stochastic differential systems and its applications in Finance. S. T. Rachev (ed.), Handbook of Computational and Numerical Methods in Finance © Springer Science+Business Media New York 2004 404 Z. Zheng • Numerical methods for stochastic differential equations (SDEs), stochastic optimal control problems and stochastic differential games.! The recognized procedures for stochastic system simulation include time discretization schemes (Euler scheme, Milshtein scheme, Runge-Kutta scheme, Romberg extrapolations) & Monte Carlo/quasi-Monte Carlo method (cf. Asmussen et al. [6], Boyle [27], Boyle et al. [28], Clark [36], Duffie and Glynn [48], Jacod and Protter [72], Kloeden and Platen [77], Kohatsu-Higa and Ogawa [79], Kohatsu-Higa and Protter [80], Kurtz and Protter [84], Milshtein [103], [104], Newton [110], [111], [112], Niederreiter [113], [114], [115], Talay [126], [127], Talay and Tubaro [129]), the Markov chain approximation method (cf. Kushner and Dupuis [87]), the lattice binomial & multinomial approximation (cf. Cox-Ross-Rubinstein [37]), the forward shooting grid method (cf. Barraquand and Pudet [19]), etc. On one hand, these methods provide probabilistic algorithms for deterministic nonlinear PDEs (cf. Kushner [86], [85], Talay and Tubaro [128], Talay and Zheng [131]), on the other hand, they provide direct approaches to simulate the sophisticated stochastic Finance models (cf. Rogers and Talay [123], Gobet and Temam [68]) and to access some quantities which are impossible or difficult to achieve via deterministic algorithms, e.g., the numerical computation of Valueat-Risk (cf. Talay and Zheng [130], [133]). An important advantage of such probabilistic algorithms is that they themselves can be interpreted as natural discrete models of the same financial situation, and thus have an inherent value independent of their use as a numerical method. In addition, as a well-known merit of the Monte Carlo method, probabilistic algorithms are less dimension-sensitive than their deterministic counterparts, which is especially important for large scale Financial Engineering. Finally we would like to point out the recent successful applications of Malliavin calculus in the numerical methods for stochastic systems (cf. Bally and Talay [13], [14], Protter and Talay [121], Fournie et al [58], [59], Gobet [66], Kohatsu-Higa [78]). • Statistical procedures and Filtering to identify model structures (cf. Bamdorff-Nielsen and Shephard [17], Del Moral et al. [105], Florens-Zmirou [56], Fournie and Talay [60], Kutoyants [91], [92], Viens [138]). • Numerical methods for backward stochastic differential equations (BSDEs). The notion of backward stochastic differential equations (BSDEs) was introduced by Pardoux and Peng [118]. Independently, Duffie and Epstein [46], [47] introduced stochastic differential utilities in economics models, as solutions to certain BSDEs . From then on, BSDEs found numerous applications in mathematical economics and finance (cf....