Discretization and simulation of stochastic differential equations. (English) Zbl 0554.60062

Let \(X_ t\) be a solution of the following ItĂ´-type stochastic differential equation \(dX_ t=b(X_ t)dt+\sigma (X_ t)dW_ t\). The authors consider approximation schemes of the type \(X_{t_{i+1}}=f(x_{t_ i},W.)\) for the processes \(X_ t\). They discuss both pathwise and mean-square convergence of several approximation schemes and estimate the speed of convergence and errors.
As far as the pathwise approximation is concerned the results are connected to the second authors paper, Stochastics 9, 275-306 (1983; Zbl 0512.60041). In the second part the authors discuss schemes for which \(\limsup_{n\to \infty}n^ 2E(| X^ n_ t-X_ t|^ 2)=C_ t\) holds and compare this with several approximations suggested in the literature. Especially the results from many doctorial theses are mentioned which are not published in regular journals. There are finally some numerical results by Monte-Carlo-approximations and stability discussions.
Reviewer: M.Breger


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C05 Monte Carlo methods
65L05 Numerical methods for initial value problems involving ordinary differential equations
93E25 Computational methods in stochastic control (MSC2010)


Zbl 0512.60041
Full Text: DOI


[1] ArnoldL.:Stochastic Differential Equations, Wiley, New-York, 1974.
[2] Ben Arous, G: ?Repr?sentation explicite de la solution de certaines ?quations diff?rentielles stochastiques?, Th?se de 3? cycle, Univ. Paris 7, 1981.
[3] ClarkJ. M. C.: ?The Design of Robust Approximation to the Stochastic Differential Equation of Nonlinear Filtering?, inCommunication Systems and Random Process Theory, J.Skwirzynskii (ed.), Sijthoff & Noordhoff, Alphen aan den Rijn, 1980.
[4] ClarkJ. M. C.: ?An Efficient Approximation Scheme for a Class of Stochastic Differential Equations?, inAdvances in Filtering and Optimal Stochastic Control, W.Fleming and L.Gorostiza (eds.), Lecture Notes in Control and Information Sciences Vol. 42, Springer-Verlag, Berlin, 1982.
[5] ClarkJ. M. C. and CameronR. J.: ?The Maximum Rate of Convergence of Discrete Approximations for Stochastic Differential Equations?, inStochastic Differential Systems, B.Grigelionis (ed.), Lecture Notes in Control and Information Sciences Vol. 25, Springer-Verlag, Berlin, 1980.
[6] DossH.: ?Liens entre ?quations diff?rentielles stochastiques et ordinaires?,Ann. Inst. H. Poincar? 13 (1977), 99.
[7] Gear, C. W.:Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, 1971. · Zbl 1145.65316
[8] Glorennec, P.: ?Estimation a priori des erreurs dans la r?solution num?rique d’?quations diff?rentielles stochastiques?, Th?se de 3? cycle, Univ. de Rennes, 1977.
[9] HenriciP.:Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, 1962.
[10] IkedaN. and WatanabeS.:Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981.
[11] KrenerA. J. and LobryC.: ?The Complexity of Stochastic Differential Equations?,Stochastics 4 (1981), 193.
[12] KunitaH.: ?On the Decomposition of Solutions of Stochastic Differential Equations?, inStochastic Integrals, D.Williams (ed.), Lecture Notes in Mathematics Vol. 851, Springer-Verlag, Berlin, 1981.
[13] Le Gland, F.: ?Estimation de param?tres dans les processus stochastiques, en observation incompl?te. Application ? un probl?me de radio-astronomie?, Th?se de Doet. Ing., Univ. Paris 9, 1981.
[14] McShaneE.J.:Stochastic Calculus and Stochastic Models, Academic Press, New York, 1974.
[15] MilshteinG. N.: ?Approximate Integration of Stochastic Differential Equations?,Theory Prob. App. 19 (1974), 557.
[16] MilshteinG. N.: ?A Method of Second Order Accuracy Integration of Stochastic Differential Equations?,Theory Prob. Appl. 23 (1976), 396. · Zbl 0422.60048
[17] Newton, N. J.: Ph D Thesis; E.E. Dept, Imperial College, London, 1983.
[18] Platen, E.: ?An Approximation Method for a Class of It? Processes?,Liet. matem. rink (1980). · Zbl 0423.60054
[19] RaoN.J., BorwankarJ.D., and RamakrishnaD.: ?Numerical Solution of It? Integral Equations?,Siam J. Control 12 (1974), 124. · Zbl 0275.65032
[20] R?melin W.: ?Numerical Treatment of Stochastic Differential Equations?,Siam J. Num. Anal. 19 (1982).
[21] SussmannH. J.: ?On the Gap Between Deterministic and Stochastic Ordinary Differential Equations?,Ann. Prob. 6 (1978), 19. · Zbl 0391.60056
[22] Talay, D.: ?Analyse num?rique des ?quations diff?rentielles stochastiques?, Th?se de 3? Cycle, Univ. de Provence (1982).
[23] TalayD.: ?R?solution trajectorielle et analyse num?rique des ?quations diff?rentielles stochastiques?,Stochastic 9 (1983), 275.
[24] TalayD.: ?Efficient Numerical Schemes for the Approximation of Expectations of Functionals of the Solution of a SDE, and Applications?, ?Filtering and Control of Random Processes?, Lecture Notes in Control and Information Sciences, Vol. 61, Springer-Verlag, Berlin, 1984.
[25] Talay, D.: ?Discr?tisation d’une EDS et calcul approch? d’esp?rances de fonctionnelles de la solution?, to appear.
[26] YamadaT.: ?Sur l’approximation des ?quations diff?rentielles stochastiques?,Zeitschift f?r Wahrschein. 36 (1976), 133.
[27] YamatoY.: ?Stochastic Differential Equations and Nilpotent Lie Algebra?,Zeitschrift f?r Wahrschein. 47 (1979), 213. · Zbl 0427.60069
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