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The Euler scheme with irregular coefficients. (English) Zbl 1020.60054

The author considers a \(d\)-dimensional system of stochastic differential equations, driven by Brownian motion \(B\) in \(R^r\), \[ X_t = X_0 + \int_0^t b(s,X_s) ds + \int_0^t \sigma(s,X_s) dB_s,\tag{1} \] where \(X_0\) is an \(R^d\)-valued random variable, independent of \(B\). For the coefficient functions he assumes necessary and sufficient conditions such that equation (1) has a unique weak solution, based on a result by H. J. Engelbert and W. Schmidt [in: Stochastic differential systems. Lect. Notes Control Inf. Sci. 69, 143-155 (1985; Zbl 0583.60052)]. He then considers the continuous Euler scheme \[ X_t^n = X_{\tau_k^n}^n + b(\tau_k^n,X_{\tau_k^n}^n)(t-\tau_k^n) + \sigma(\tau_k^n,X_{\tau_k^n}^n)(B_t - B_{\tau_k^n}),\tag{2} \] for \({\tau_k^n} \leq t \tau_{k+1}^n, k=0,1,\ldots,n\), where \(0=\tau_0^n \leq \tau_1^n \leq \ldots \leq \tau_n^n = T\) is a sequence of random partitions of \([0,T]\). The aim of the paper is to show weak convergence of the solution of the continuous Euler scheme (2) to the unique weak solution of (1) under only mild smoothness conditions on the coefficient functions \(b\) and \(\sigma\). Under additional conditions the author proves results concerning the rate of convergence in the one-dimensional case.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C05 Monte Carlo methods
60F05 Central limit and other weak theorems

Citations:

Zbl 0583.60052
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References:

[1] BALLY, V. and TALAY, D. (1995). The Euler scheme for stochastic differential equations: error analysis with Malliavin calculus. Math. Comput. Simulation 38 35-41. · Zbl 0824.60056 · doi:10.1016/0378-4754(93)E0064-C
[2] BALLY, V. and TALAY, D. (1996). The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Related Fields 104 43-60. · Zbl 0925.60064 · doi:10.1002/zamm.19960761107
[3] BALLY, V. and TALAY, D. (1996). The law of the Euler scheme for stochastic differential equations. II. Convergence rate of the density. Monte Carlo Methods Appl. 2 93-128. · Zbl 0925.60064 · doi:10.1002/zamm.19960761107
[4] BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[5] BLUMENTHAL, R. M. and GETOOR, R. K. (1968). Markov Processes and Potential Theory. Academic Press, New York. · Zbl 0169.49204
[6] BOy LE, P., BROADIE, M. and GLASSERMAN, P. (1997). Monte Carlo methods for security pricing. Computational financial modelling. J. Econom. Dy nam. Control 21 1267-1321. · Zbl 0901.90007 · doi:10.1016/S0165-1889(97)00028-6
[7] CHAN, K. S. and STRAMER, O. (1998). Weak convergency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients. Stochastic Process. Appl. 76 33-44. · Zbl 0934.60052 · doi:10.1016/S0304-4149(98)00020-9
[8] CHORIN, A.J. (1977). Hermite expansions in Monte Carlo computation. J. Comput. Phy s. 8 472-482. · Zbl 0229.65025 · doi:10.1016/0021-9991(71)90025-8
[9] DUFFIE, D. and GLy NN, P. (1995). Efficient Monte Carlo simulation of security prices. Ann. Appl. Probab. 5 879-905. · Zbl 0877.65099 · doi:10.1214/aoap/1177004598
[10] ENGELBERT, H. J. and SCHMIDT, W. (1984). On one-dimensional stochastic differential equations with generalized drift. Stochastic Differential Sy stems. Lecture Notes in Control and Inform. Sci. 69 143-155. Springer, Berlin. · Zbl 0583.60052
[11] IKEDA, N. and WATANABE, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam. · Zbl 0495.60005
[12] JACOD, J. and PROTTER, P. (1998). Asy mptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267-307. · Zbl 0937.60060 · doi:10.1214/aop/1022855419
[13] KARATZAS, I. and SHREVE, S. (1988). Brownian Motion and Stochastic Calculus. Springer, New York. · Zbl 0638.60065
[14] KLOEDEN, P. E. and PLATEN, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin. · Zbl 0752.60043
[15] KOHATSU-HIGA, A. and PROTTER, P. (1994). The Euler scheme for SDEs driven by semimartingales. In Stochastic Analy sis on Infinite Dimensional Spaces (H. Kunita and H.-H. Kuo, eds.) 141-151. Longman, Harlow, UK. · Zbl 0814.65142
[16] KURTZ, T. G. and PROTTER, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035-1070. · Zbl 0742.60053 · doi:10.1214/aop/1176990334
[17] KURTZ, T. G. and PROTTER, P. (1996). Weak Convergence of stochastic integrals and differential equations. Probabilistic Models for Nonlinear Partial Differential Equations. Lecture Notes in Math. 1627 1-41. Springer, Berlin. · Zbl 0862.60041 · doi:10.1007/BFb0093176
[18] NEWTON, N. J. (1994). Variance reduction for simulated diffusions. SIAM J. Appl. Math. 54 1780-1805. JSTOR: · Zbl 0811.60046 · doi:10.1137/S0036139992236220
[19] PROTTER, P. (1990). Stochastic Integrations and Differential Equations. A New Approach. Springer, New York. · Zbl 0694.60047
[20] PROTTER, P. and TALAY, D. (1997). The Euler Scheme for Lévy-driven stochastic differential equations. Ann. Probab. 25 393-423. · Zbl 0876.60030 · doi:10.1214/aop/1024404293
[21] REVUZ, D. and YOR, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin. · Zbl 0731.60002
[22] TALAY, D. (1995). Simulation and numerical analysis of stochastic differential sy stems: a review. Probabilistic Methods in Applied physics. Lecture Notes in Phy s. 451 63-106. Springer, Berlin.
[23] TALAY, D. and TUBARO, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl. 8 94-120. · Zbl 0718.60058 · doi:10.1080/07362999008809220
[24] TANEÉ, E. (2001). Probabilistic study of Smoluchowski’s equations; Euler schemes for functionals; amplitude of Brownian motions with drift. Ph.D. thesis, Univ. Nancy. Available at www.iecn.u-nancy.fr/ tanre/. URL:
[25] VAN DER VAART, A. W. and WELLNER, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002
[26] WAGNER, W. (1998). Monte Carlo evaluation of functionals of solutions of stochastic differential equations. Variance reduction and numerical examples. Stochastic Anal. Appl. 6 447-468. · Zbl 0671.60058 · doi:10.1080/07362998808809161
[27] VANCOUVER, BC CANADA V6T 1Z2 E-MAIL: ly an@pims.math.ca
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