Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-28T02:02:14.316Z Has data issue: false hasContentIssue false
  • English
  • Français

An introduction to numerical methods for stochastic differential equations

Published online by Cambridge University Press:  07 November 2008

Eckhard Platen
Eckhard Platen
Affiliation:
School of Mathematical Sciences and School of Finance and Economics, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper aims to give an overview and summary of numerical methods for the solution of stochastic differential equations. It covers discrete time strong and weak approximation methods that are suitable for different applications. A range of approaches and results is discussed within a unified framework. On the one hand, these methods can be interpreted as generalizing the well-developed theory on numerical analysis for deterministic ordinary differential equations. On the other hand they highlight the specific stochastic nature of the equations. In some cases these methods lead to completely new and challenging problems.

Type
Research Article
Information
Acta Numerica , Volume 8 , January 1999 , pp. 197 - 246
Copyright
Copyright © Cambridge University Press 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abukhaled, M. I. and Allen, E. J. (1998), ‘A recursive integration method for approximate solution of stochastic differential equations’, Intern. J. Comput. Math. 66, 53–66.CrossRefGoogle Scholar
Allain, M. F. (1974), Sur quelques types d'approximation des solutions d'équations différentielles stochastiques, PhD thesis, Univ. Rennes.Google Scholar
Allen, E. J., Novosel, S. J. and Zhang, Z. (1998), ‘Finite element and difference approximation of some linear stochastic partial differential equations’, Stochastics and Stochastics Reports 64, 117–142.CrossRefGoogle Scholar
Anderson, S. L. (1990), ‘Random number generators on vector supercomputers and other advanced structures’, SIAM Review 32, 221–251.CrossRefGoogle Scholar
Antipov, M. V. (1995), ‘Congruence operator of the pseudo-random numbers generator and a modification of Euclidean decomposition’, Monte Carlo Methods Appl. 1, 203–219.CrossRefGoogle Scholar
Antipov, M. V. (1996), ‘Sequences of numbers for Monte Carlo methods’, Monte Carlo Methods Appl. 2, 219–235.CrossRefGoogle Scholar
Arnold, L. (1974), Stochastic Differential Equations, Wiley, New York.Google Scholar
Arnold, L. and Kloeden, P. E. (1996), ‘Discretization of a random dynamical system near a hyperbolic point’, Mathematische Nachrichten 181, 43–72.CrossRefGoogle Scholar
Artemiev, S. S. (1985), ‘A variable step algorithm for numerical solution of stochastic differential equations’, Chisl. Metody Mekh. Sploshn. Sredy 16, 11–23. In Russian.Google Scholar
Artemiev, S. S. (1993 a), Certain aspects of application of numerical methods of solving SDE systems, in Numer. Anal., Vol. 1 of Bulletin of the Novosibirsk Computing Center, NCC Publisher, pp. 1–16.Google Scholar
Artemiev, S. S. (1993 b), The stability of numerical methods for solving stochastic differential equations, in Numer. Anal., Vol. 2 of Bulletin of the Novosibirsk Computing Center, NCC Publisher, pp. 1–10.Google Scholar
Artemiev, S. S. (1994), ‘The mean square stability of numerical methods for solving stochastic differential equations’, Russ. J. Numer. Anal. Math. Model. 9, 405–416.Google Scholar
Artemiev, S. S. and Averina, A. T. (1997), Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations, VSP, Utrecht.CrossRefGoogle Scholar
Artemiev, S. S. and Shkurko, I. O. (1991), ‘Numerical analysis of dynamics of oscillatory stochastic systems’, Soviet J. Numer. Anal. Math. Model. 6, 277–298.Google Scholar
Asmussen, S., Glynn, P. and Pitman, J. (1995), ‘Discretization error in simulation of one-dimensional reflecting Brownian motion’, Ann. Appl. Probab. 5, 875–896.CrossRefGoogle Scholar
Atalla, M. A. (1986), Finite-difference approximations for stochastic differential equations, in Probabilistic Methods for the Investigation of Systems with an Infinite Number of Degrees of Freedom, Collection of Scientific Works, Kiev, pp. 11–16. In Russian.Google Scholar
Averina, A. T. and Artemiev, S. S. (1986), ‘A new family of numerical methods for solving stochastic differential equations’, Soviet. Math. Dokl. 33, 736–738.Google Scholar
Averina, A. T. and Artemiev, S. S. (1988), ‘Numerical solutions of systems of stochastic differential equations’, Soviet J. Numer. Anal. Math. Model. 3, 267–285.Google Scholar
Azencott, R. (1982), Stochastic Taylor formula and asymptotic expansion of Feynman integrals, in Séminaire de probabilités XVI, Supplement, Vol. 921 of Lecture Notes in Math., Springer, pp. 237–285.Google Scholar
Bachelier, L. (1900), ‘Théorie de la spéculation’, Annales de l'Ecole Normale Supérieure, Series 3 17, 21–86.CrossRefGoogle Scholar
Bally, V. (1989 a), ‘Approximation for the solution of stochastic differential equations. I: Lp-convergence’, Stochastics and Stochastics Reports 28, 209–246.CrossRefGoogle Scholar
Bally, V. (1989 b), ‘Approximation for the solution of stochastic differential equations. II: Strong-convergence’, Stochastics and Stochastics Reports 28, 357–385.CrossRefGoogle Scholar
Bally, V. (1990), ‘Approximation for the solutions of stochastic differential equations. III: Jointly weak convergence’, Stochastics and Stochastics Reports 30, 171–191.CrossRefGoogle Scholar
Bally, V. and Talay, D. (1995), ‘The Euler scheme for stochastic differential equations: Error analysis with Malliavin calculus’, Math. Comput. Simul. 38, 35–41.CrossRefGoogle Scholar
Bally, V. and Talay, D. (1996 a), ‘The law of the Euler scheme for stochastic differential equations I: Convergence rate of the distribution function’, Probability Theory Related Fields 104, 43–60.CrossRefGoogle Scholar
Bally, V. and Talay, D. (1996 b), ‘The law of the Euler scheme for stochastic differential equations II: Convergence rate of the density function’, Monte Carlo Methods Appl. 2, 93–128.CrossRefGoogle Scholar
Barraquand, J. (1995), ‘Monte Carlo integration, quadratic resampling, and asset pricing’, Math. Comput. Simul. 38, 173–182.CrossRefGoogle Scholar
BenArous, G. (1989), ‘Flots et series de Taylor stochastiques’, Probability Theory Related Fields 81, 29–77.Google Scholar
Bensoussan, A., Glowinski, R. and Rascanu, A. (1990), ‘Approximation of the Zakai equation by the splitting up method’, SIAM J. Control Optimiz. 28, 1420–1431.CrossRefGoogle Scholar
Bensoussan, A., Glowinski, R. and Rascanu, A. (1992), ‘Approximation of some stochastic differential equations by the splitting up method’, Appl. Math. Optim. 25, 81–106.CrossRefGoogle Scholar
Björck, A. and Dahlquist, G. (1974), Numerical Methods. Series in Automatic Computation, Prentice-Hall, New York.Google Scholar
Black, F. and Scholes, M. (1973), ‘The pricing of options and corporate liabilities’, J. Political Economy 81, 637–659.CrossRefGoogle Scholar
Bouleau, N. (1990), ‘On effective computation of expectations in large or infinite dimension: Random numbers and simulation’, J. Comput. Appl. Math. 31, 23–34.CrossRefGoogle Scholar
Bouleau, N. and Lépingle, D. (1993), Numerical Methods for Stochastic Processes, Wiley, New York.Google Scholar
Box, G. and Muller, M. (1958), ‘A note on the generation of random normal variables’, Ann. Math. Statist. 29, 610–611.CrossRefGoogle Scholar
Boyce, W. E. (1978), ‘Approximate solution of random ordinary differential equations’, Adv. Appl. Probab. 10, 172–184.CrossRefGoogle Scholar
Boyle, P. P. (1977), ‘A Monte Carlo approach’, J. Financial Economics 4, 323–338.CrossRefGoogle Scholar
Bratley, P., Fox, B. L. and Schrage, L. (1987), A Guide to Simulation, 2nd edn, Springer, New York.CrossRefGoogle Scholar
Brent, R. P. (1974), ‘A Gaussian pseudo number generator’, Commun. Assoc. Comput. Mach. 17, 704–706.Google Scholar
Burrage, K. (1995), Parallel and Sequential Methods for Ordinary Differential Equations, Clarendon Press, Oxford University Press.CrossRefGoogle Scholar
Burrage, K. and Burrage, P. M. (1996), ‘High strong order explicit Runge–Kutta methods for stochastic ordinary differential equations’, Appl. Numer. Math. 22, 81–101.CrossRefGoogle Scholar
Burrage, K. and Burrage, P. M. (1998), ‘General order conditions for stochastic Runge–Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems’, Appl. Numer. Math. 28, 161–177.CrossRefGoogle Scholar
Burrage, K. and Platen, E. (1994), ‘Runge–Kutta methods for stochastic differential equations’, Ann. Numer. Math. 1, 63–78.Google Scholar
Burrage, K., Burrage, P. M. and Belward, J. A. (1997), ‘A bound on the maximum strong order of stochastic Runge–Kutta methods for stochastic ordinary differential equations’, BIT 37, 771–780.CrossRefGoogle Scholar
Burrage, P. M. (1998), Runge–Kutta methods for stochastic differential equations, PhD thesis, University of Queensland, Brisbane, Australia.Google Scholar
Butcher, J. C. (1987), The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods, Wiley, Chichester.Google Scholar
Cambanis, S. and Hu, Y. Z. (1996), ‘Exact convergence rate of the Euler–Maruyama scheme and application to sample design’, Stochastics and Stochastics Reports 59, 211–240.CrossRefGoogle Scholar
Casasus, L. L. (1982), On the numerical solution of stochastic differential equations and applications, in Proceedings of the Ninth Spanish–Portuguese Conference on Mathematics, Vol. 46 of Acta Salmanticensia Ciencias, Univ. Salamanca, pp. 811–814. In Spanish.Google Scholar
Casasus, L. L. (1984), On the convergence of numerical methods for stochastic differential equations, in Proceedings of the Fifth Congress on Differential Equations and Applications, Univ. La Laguna, pp. 493–501. Puerto de la Cruz (1982), in Spanish, Informes 14.Google Scholar
Castell, F. and Gaines, J. (1995), ‘An efficient approximation method for stochastic differential equations by means of the exponential Lie series’, Math. Comput. Simul. 38, 13–19.CrossRefGoogle Scholar
Castell, F. and Gaines, J. (1996), ‘The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations’, Ann. Inst. H. Poincaré Probab. Statist. 32, 231–250.Google Scholar
Chan, K. S. and Stramer, O. (1998), ‘Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients’, Stochastic Process. Appl. 76, 33–44.CrossRefGoogle Scholar
Chang, C. C. (1987), ‘Numerical solution of stochastic differential equations with constant diffusion coefficients’, Math. Comput. 49, 523–542.CrossRefGoogle Scholar
Chevance, D. (1997), Numerical methods for backward stochastic differential equations, in Numerical Methods in Finance (Rogers, L. C. G. and Talay, D., eds), Cambridge University Press, pp. 232–244.CrossRefGoogle Scholar
Clark, J. M. C. (1978), The design of robust approximations to the stochastic differential equations of nonlinear filtering, in Communication Systems and Random Processes Theory (Skwirzynski, J. K., ed.), Vol. 25 of NATO ASI Series E: Applied Sciences, Sijthoff and Noordhoff, Alphen aan den Rijn, pp. 721–734.CrossRefGoogle Scholar
Clark, J. M. C. (1982 a), An efficient approximation scheme for a class of stochastic differential equations, in Advances in Filtering and Optimal Stochastic Control, Vol. 42 of Lecture Notes in Control and Inform. Sci., Springer, pp. 69–78.CrossRefGoogle Scholar
Clark, J. M. C. (1982 b), A nice discretization for stochastic line integrals, in Stochastic Differential Systems, Vol. 69 of Lecture Notes in Control and Inform. Sci., Springer, pp. 131–142.Google Scholar
Clark, J. M. C. and Cameron, R. J. (1980), The maximum rate of convergence of discrete approximations for stochastic differential equations, in Stochastic Differential Systems (Grigelionis, B., ed.), Vol. 25 of Lecture Notes in Control and Inform. Sci., Springer, pp. 162–171.Google Scholar
Clements, D. J. and Anderson, B. D. O. (1973), ‘Well behaved Itô equations with simulations that always misbehave’, IEEE Trans. Automat. Control 18, 676–677.CrossRefGoogle Scholar
Cyganowski, S. O. (1995), A Maple package for stochastic differential equations, in Computational Techniques and Applications: CTAC95 (Easton, A. K. and May, R. L., eds), World Scientific.Google Scholar
Cyganowski, S. O. (1996), ‘Solving stochastic differential equations with Maple’, Maple Tech. 3, 38.Google Scholar
Dashevski, M. I. and Liptser, R. S. (1966), ‘Simulation of stochastic differential equations connected with the disorder problem by means of analog computer’, Autom. Remote Control 27, 665–673. In Russian.Google Scholar
Denk, G. and Schäffer, S. (1997), ‘Adam's methods for the efficient solution of stochastic differential equations with additive noise’, Computing 59, 153–161.CrossRefGoogle Scholar
Douglas, J., Ma, J. and Protter, P. (1996), ‘Numerical methods for forward-backward stochastic differential equations’, Ann. Appl. Probab. 6, 940–968.CrossRefGoogle Scholar
Drummond, I. T., Duane, S. and Horgan, R. R. (1983), ‘The stochastic method for numerical simulations: Higher order corrections’, Nuc. Phys. B220 FS8, 119–136.CrossRefGoogle Scholar
Drummond, I. T., Hoch, A. and Horgan, R. R. (1986), ‘Numerical integration of stochastic differential equations with variable diffusivity’, J. Phys. A: Math. Gen. 19, 3871–3881.CrossRefGoogle Scholar
Drummond, P. D. and Mortimer, I. K. (1991), ‘Computer simulation of multiplicative stochastic differential equations’, J. Comput. Phys. 93, 144–170.CrossRefGoogle Scholar
Dsagnidse, A. A. and Tschitashvili, R. J. (1975), Approximate integration of stochastic differential equations, Tbilisi State, University, Inst. Appl. Math. ‘Trudy IV’, Tbilisi, pp. 267–279. In Russian.Google Scholar
Eichenauer, J. and Lehn, J. (1986), ‘A non-linear congruential pseudo random number generator’, Statist. Paper 27, 315–326.Google Scholar
Einstein, A. (1906), ‘Zur Theorie der Brownschen Bewegung’, Ann. Phys. IV 19, 371.CrossRefGoogle Scholar
Elliott, R. J. (1982), Stochastic Calculus and Applications, Springer.Google Scholar
Elliott, R. J. and Glowinski, R. (1989), ‘Approximations to solutions of the Zakai filtering equation’, Stoch. Anal. Appl. 7, 145–168.CrossRefGoogle Scholar
Entacher, K., Uhl, A. and Wegenkittl, S. (1998), ‘Linear congruential generators for parallel Monte Carlo: the leap-frog case’, Monte Carlo Methods Appl. 4, 1–16.CrossRefGoogle Scholar
Ermakov, S. M. (1975), Die Monte-Carlo-Methode und verwandte Fragen, Hochschulbücher für Mathematik, Band 72, VEB Deutscher Verlag der Wissenschaften, Berlin. In German: translation from Russian by E. Schincke and M. Schleiff.Google Scholar
Ermakov, S. M. and , Mikhailov (1982), Statistical Modeling, 2nd edn, Nauka, Moscow.Google Scholar
Fahrmeier, L. (1974), ‘Schwache Konvergenz gegen Diffusionsprozesse’, Z. Angew. Math. Mech. 54, 245.CrossRefGoogle Scholar
Fahrmeier, L. (1976), ‘Approximation von stochastischen Differenzialgleichungen auf Digital- und Hybridrechnern’, Computing 16, 359–371.CrossRefGoogle Scholar
Feng, J. F. (1990), ‘Numerical solution of stochastic differential equations’, Chinese J. Numer. Appl. 12, 28–41.Google Scholar
Feng, J. F., Lei, G. Y. and Qian, M. P. (1992), ‘Second order methods for solving stochastic differential equations’, J. Comput. Math. 10, 376–387.Google Scholar
Fischer, P. and Platen, E. (1998), Applications of the balanced method to stochastic differential equations in filtering, Technical report FMRR 005–98, Australian National University, Canberra, Financial Mathematics Research Reports.Google Scholar
Fishman, G. S. (1992), Monte Carlo: Concepts, Algorithms and Applications. Series in Operations Research, Springer.Google Scholar
Fournie, E., Lebuchoux, J. and Touzi, N. (1997), ‘Small noise expansion and importance sampling’, Asympt. Anal. 14, 331–376.Google Scholar
Fox, R. F. (1991), ‘Second-order algorithm for the numerical integration of colored-noise problems’, Phys. Rev. A 43, 2649–2654.CrossRefGoogle ScholarPubMed
Franklin, J. N. (1965), ‘Difference methods for stochastic ordinary differential equations’, Math. Comput. 19, 552–561.CrossRefGoogle Scholar
Gaines, J. G. (1994), ‘The algebra of iterated stochastic integrals’, Stochastics and Stochastics Reports 49, 169–179.CrossRefGoogle Scholar
Gaines, J. G. (1995 a), ‘A basis for iterated stochastic integrals’, Math. Comput. Simul. 38, 7–11.CrossRefGoogle Scholar
Gaines, J. G. (1995 b), Numerical experiments with S(P)DE's, in Proceedings of the ICMS Conference March 1994, Cambridge University Press.Google Scholar
Gaines, J. G. and Lyons, T. J. (1994), ‘Random generation of stochastic area integrals’, SIAM J. Appl. Math. 54, 1132–1146.CrossRefGoogle Scholar
Gaines, J. G. and Lyons, T. J. (1997), ‘Variable step size control in the numerical solution of stochastic differential equations’, SIAM J. Appl. Math. 57, 1455–1484.CrossRefGoogle Scholar
Gard, T. C. (1988), Introduction to Stochastic Differential Equations, Marcel Dekker, New York.Google Scholar
Gear, C. W. (1971), Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Gelbrich, M. (1995), ‘Simultaneous time and chance discretization for stochastic differential equations’, J. Comput. Appl. Math. 58, 255–289.CrossRefGoogle Scholar
Gelbrich, M. and Rachev, S. T. (1996), Discretization for stochastic differential equations, Lp Wasserstein metrics, and econometrical models, in Distributions with fixed marginals and related topics, Vol. 28 of IMS Lecture Notes Monogr. Ser., Inst. Math. Statist., Hayward, CA, pp. 97–119.CrossRefGoogle Scholar
Gentle, J. E. (1998), Random Number Generation and Monte Carlo Methods. Series in Statistics and Computing, Springer.CrossRefGoogle Scholar
Gerardi, A., Marchetti, F. and Rosa, A. M. (1984), ‘Simulation of diffusions with boundary conditions’, Systems Control Lett. 4, 253.CrossRefGoogle Scholar
Gikhman, I. I. and Skorokhod, A. V. (1979), The Theory of Stochastic Processes, Vol. I–III, Springer.Google Scholar
Gladyshev, S. A. and Milstein, G. N. (1984), ‘The Runge–Kutta method for calculation of Wiener integrals of functionals of exponential type’, Zh. Vychisl. Mat. Mat. Fiz 24, 1136–1149. In Russian.Google Scholar
Glorennec, P. Y. (1977), ‘Estimation a priori des erreurs dans la résolution numérique d'équations différentielles stochastiques’, Séminaire de Probabilités, Univ. Rennes 1, 57–93.Google Scholar
Glynn, P. W. and Iglehart, O. L. (1989), ‘Importance sampling for stochastic simulations’, Management Science 35, 1367–1392.CrossRefGoogle Scholar
Golec, J. (1995), ‘Stochastic averaging principle for systems with pathwise uniqueness’, Stoch. Anal. Appl. 13, 307–322.CrossRefGoogle Scholar
Golec, J. (1997), ‘Averaging Euler-type difference schemes’, Stoch. Anal. Appl. 15, 751–758.CrossRefGoogle Scholar
Golec, J. and Ladde, G. S. (1989), ‘Euler-type approximation for systems of stochastic differential equations’, J. Appl. Math. Simul. 2, 239–249.CrossRefGoogle Scholar
Golec, J. and Ladde, G. S. (1990), ‘Averaging principle and systems of singularly perturbed stochastic differential equations’, J. Math. Phys. 31, 1116–1123.CrossRefGoogle Scholar
Goodlett, S. T. and Allen, E. J. (1994), ‘A variance reduction technique for use with the extrapolated Euler method for numerical solution of stochastic differential equations’, Stoch. Anal. Appl. 12, 131–140.CrossRefGoogle Scholar
Gorostiza, L. G. (1980), ‘Rate of convergence of an approximate solution of stochastic differential equations’, Stochastics 3, 267–276. Erratum in Stochastics 4 (1981), 85.CrossRefGoogle Scholar
Grecksch, W. and Kloeden, P. E. (1996), ‘Time-discretised Galerkin approximations of parabolic stochastic PDEs’, Bull. Austral. Math. Soc. 54, 79–85.CrossRefGoogle Scholar
Grecksch, W. and Wadewitz, A. (1996), ‘Approximation of solutions of stochastic differential equations by discontinuous Galerkin methods’, J. Anal. Appl. 15, 901–916.Google Scholar
Greenside, H. S. and Helfand, E. (1981), ‘Numerical integration of stochastic differential equations’, Bell Syst. Techn. J. 60, 1927–1940.CrossRefGoogle Scholar
Greiner, A., Strittmatter, W. and Honerkamp, J. (1987), ‘Numerical integration of stochastic differential equations’, J. Statist. Phys. 51, 95–108.CrossRefGoogle Scholar
Grorud, A. and Talay, D. (1990), Approximation of Lyapunov exponents of stochastic differential systems on compact manifolds, in Analysis and Optimization of Systems, Vol. 144 of Lecture Notes in Control and Inform. Sci., Springer, pp. 704–713.CrossRefGoogle Scholar
Grorud, A. and Talay, D. (1996), ‘Approximation of Lyapunov exponents of nonlinear stochastic differential equations’, SIAM J. Appl. Math. 56, 627–650.CrossRefGoogle Scholar
Guo, S. J. (1982), ‘On the mollifier approximation for solutions of stochastic differential equations’, J. Math. Kyoto Univ. 22, 243–254.Google Scholar
Guo, S. J. (1984), ‘Approximation theorems based on random partitions for stochastic differential equations and applications’, Chinese Ann. Math. 5, 169–183.Google Scholar
Gyöngy, I. (1991), ‘On approximation of Itô stochastic equations’, Math. SSR Sbornik 70, 165–173.Google Scholar
Gyöngy, I. and Nuarlart, D. (1997), ‘Implicit scheme for stochastic partial differential equations driven by space-time white noise’, Potential Analysis 7, 725–757.CrossRefGoogle Scholar
Hairer, E. and Wanner, G. (1991), Solving ordinary differential equations II: Stiff and differential algebraic systems, Springer.CrossRefGoogle Scholar
Hairer, E., Nørsett, S. P. and Wanner, G. (1987), Solving ordinary differential equations I: Nonstiff problems, Springer.CrossRefGoogle Scholar
Halton, J. H. (1960), ‘On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals’, Numer. Math. 2, 84–90.CrossRefGoogle Scholar
Hammersley, J. M. and Handscomb, D. C. (1964), Monte Carlo Methods, Methuen, London.CrossRefGoogle Scholar
Harris, C. J. (1976), Simulation of nonlinear stochastic equations with applications in modelling water pollution, in Mathematical Models for Environmental Problems (Brebbi, C. A., ed.), Pentech Press, London, pp. 269–282.Google Scholar
Hausenblas, E. (1999 a), A Monte-Carlo method with inherited parallelism for solving partial differential equations with boundary conditions numerically, Dept. Math., University of Salzburg, Austria. Paper in progress.Google Scholar
Hausenblas, E. (1999 b), A numerical scheme using excursion theory for simulating stochastic differential equations with reflection and local time at a boundary, Dept. Math., University of Salzburg, Austria. Paper in progress.Google Scholar
Haworth, D. C. and Pope, S. B. (1986), ‘A second-order Monte-Carlo method for the solution of the Itô stochastic differential equation’, Stoch. Anal. Appl. 4, 151–186.CrossRefGoogle Scholar
Heath, D. and Platen, E. (1996), ‘Valuation of FX barrier options under stochastic volatility’, Financial Engineering and the Japanese Markets 3, 195–215.CrossRefGoogle Scholar
Helfand, E. (1979), ‘Numerical integration of stochastic differential equations’, Bell Syst. Techn. J. 58, 2289–2299.CrossRefGoogle Scholar
Hernandez, D. B. and Spigler, R. (1992), ‘A-stability of implicit Runge–Kutta methods for systems with additive noise’, BIT 32, 620–633.CrossRefGoogle Scholar
Hernandez, D. B. and Spigler, R. (1993), ‘Convergence and stability of implicit Runge–Kutta methods for systems with multiplicative noise’, BIT 33, 654–669.CrossRefGoogle Scholar
Higham, D. J. (1998), Mean-square and asymptotic stability of numerical methods for stochastic ordinary differential equations, Strathclyde Mathematics Research Report 39, University of Strathclyde, Glasgow, UK.Google Scholar
Hofmann, N. (1994), Beiträge zur schwachen Approximation stochastischer Differentialgleichungen, PhD thesis, Dissertation Humboldt Universität Berlin.Google Scholar
Hofmann, N. (1995), ‘Stability of weak numerical schemes for stochastic differential equations’, Math. Comput. Simul. 38, 63–68.CrossRefGoogle Scholar
Hofmann, N. and Mathé, P. (1997), ‘On quasi-Monte Carlo simulation of stochastic differential equations’, Math. Comput. 66, 573–589.CrossRefGoogle Scholar
Hofmann, N. and Platen, E. (1994), ‘Stability of weak numerical schemes for stochastic differential equations’, Comput. Math. Appl. 28, 45–57.CrossRefGoogle Scholar
Hofmann, N. and Platen, E. (1996), ‘Stability of superimplicit numerical methods for stochastic differential equations’, Fields Institute Communications 9, 93–104.Google Scholar
Hofmann, N., Müller-Gronbach, T. and Ritter, K. (1998), Optimal approximation of stochastic differential equations by adaptive step-size control, Preprint Nr. A-9-98, Fachbereich Mathematik, Freie Universität Berlin.Google Scholar
Hofmann, N., Platen, E. and Schweizer, M. (1992), ‘Option pricing under incompleteness and stochastic volatility’, Mathematical Finance 2, 153–187.CrossRefGoogle Scholar
Hu, Y. Z. (1992), Series de Taylor stochastique et formule de Campbell-Hausdorff d'après Ben Arous, in Séminaire de Probabilités XXVI, Vol. 1526 of Lecture Notes in Math., Springer, pp. 587–594.CrossRefGoogle Scholar
Hu, Y. Z. (1996), Strong and weak order of time discretization schemes of stochastic differential equations, in Séminaire de Probabilités XXX, Vol. 1626 of Lecture Notes in Math., Springer, pp. 218–227.CrossRefGoogle Scholar
Hu, Y. Z. and Meyer, P. A. (1993), ‘On the approximation of multiple Stratonovich integrals’, in Stochastic Processes, Springer, pp. 141–147.CrossRefGoogle Scholar
Hu, Y. Z. and Watanabe, S. (1996), ‘Donsker's delta functions and approximation of heat kernels by the time discretization methods’, J. Math. Kyoto Univ. 36, 499–518.Google Scholar
Hull, J. and White, A. (1988), ‘The use of control variate techniques in option pricing’, J. Financial and Quantitative Analysis 23, 237–251.CrossRefGoogle Scholar
Ikeda, N. and Watanabe, S. (1989), Stochastic Differential Equations and Diffusion Processes, 2nd edn, North-Holland, Amsterdam. (1st edn (1981).)Google Scholar
Itô, K. (1944), ‘Stochastic integral’, Proc. Imp. Acad. Tokyo 20, 519–524.Google Scholar
Jacod, J. and Protter, P. (1998), ‘Asymptotic error distribution for the Euler method for stochastic differential equations’, Ann. Probab. 26, 267–307.CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (1987), Limit Theorems for Stochastic Processes, Springer.CrossRefGoogle Scholar
Janicki, A. (1996), Numerical and Statistical Approximation of Stochastic Differential Equations with Non-Gaussian Measures, H. Steinhaus Center for Stochastic Methods in Science and Technology, Wroclaw, Poland.Google Scholar
Janicki, A. and Weron, A. (1994), Simulation of Chaotic Behavior of α-stable Stochastic Processes, Vol. 178 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York.Google Scholar
Janicki, A., Michna, Z. and Weron, A. (1996), ‘Approximation of stochastic differential equations driven by α-stable Lévy motion’, Applicationes Mathematicae 24, 149–168.CrossRefGoogle Scholar
Janssen, R. (1984 a), ‘Difference-methods for stochastic differential equations with discontinuous coefficients’, Stochastics 13, 199–212.CrossRefGoogle Scholar
Janssen, R. (1984 b), ‘Discretization of the Wiener process in difference methods for stochastic differential equations’, Stochastic Process. Appl. 18, 361–369.CrossRefGoogle Scholar
Joy, C., Boyle, P. P. and Tan, K. S. (1996), ‘Quasi Monte Carlo methods in numerical finance’, Management Science 42, 926–938.CrossRefGoogle Scholar
Kalos, M. H. and Whitlock, P. A. (1986), Monte Carlo Methods, Wiley-Interscience, New York.CrossRefGoogle Scholar
Kanagawa, S. (1988), ‘The rate of convergence for Maruyama's approximate solutions of stochastic differential equations’, Yokohama Math. J. 36, 79–85.Google Scholar
Kanagawa, S. (1989), ‘The rate of convergence for approximate solutions of stochastic differential equations’, Tokyo J. Math. 12, 33–48.CrossRefGoogle Scholar
Kanagawa, S. (1995), ‘Error estimation for the Euler–Maruyama approximate solutions of stochastic differential equations’, Monte Carlo Methods Appl. 1, 165–171.CrossRefGoogle Scholar
Kanagawa, S. (1996), Convergence rates for the Euler–Maruyama type approximate solutions of stochastic differential equations, in Probability Theory and Mathematical Statistics, Proceedings of the Seventh Japan–Russia Symposium, World Scientific, Singapore, pp. 183–192.Google Scholar
Kanagawa, S. (1997), ‘Confidence intervals of discretized Euler–Maruyama approximate solutions of SDE's’, Nonlinear Analysis, Theory, Methods and Applications, Vol. 30, pp. 4101–4103.Google Scholar
Kaneko, T. and Nakao, S. (1988), A note on approximations for stochastic differential equations, in Séminaire de Probabilités XXII, Vol. 1321 of Lecture Notes in Math., Springer, pp. 155–162.CrossRefGoogle Scholar
Kannan, D. and Wu, D. T. (1993), ‘A numerical study of the additive functionals of solutions of stochastic differential equations’, Dyn. Sys. Appl. 2, 291–310.Google Scholar
Karatzas, I. and Shreve, S. E. (1988), Brownian Motion and Stochastic Calculus, Springer.CrossRefGoogle Scholar
Kendall, W. S. (1993), Doing stochastic calculus with Mathematica, in Economic and financial modeling with Mathematica, TELOS, Santa Clara, CA, pp. 214–238.CrossRefGoogle Scholar
Klauder, J. R. and Petersen, W. P. (1985), ‘Numerical integration of multiplicative-noise stochastic differential equations’, SIAM J. Numer. Anal. 6, 1153–1166.CrossRefGoogle Scholar
Kloeden, P. E. and Pearson, R. A. (1977), ‘The numerical solution of stochastic differential equations’, J. Austral. Math. Soc. Ser. B 20, 8–12.CrossRefGoogle Scholar
Kloeden, P. E. and Platen, E. (1989), ‘A survey of numerical methods for stochastic differential equations’, J. Stochastic Hydrology and Hydraulics 3, 155–178.CrossRefGoogle Scholar
Kloeden, P. E. and Platen, E. (1991 a), ‘Relations between multiple Itô and Stratonovich integrals’, Stoch. Anal. Appl. IX, 86–96.Google Scholar
Kloeden, P. E. and Platen, E. (1991 b), ‘Stratonovich and Itô stochastic Taylor expansions’, Mathematische Nachrichten 151, 33–50.CrossRefGoogle Scholar
Kloeden, P. E. and Platen, E. (1992), ‘Higher order implicit strong numerical schemes for stochastic differential equations’, J. Statist. Phys. 66, 283–314.CrossRefGoogle Scholar
Kloeden, P. E. and Platen, E. (1992/1995 b), Numerical Solution of Stochastic Differential Equations, Vol. 23 of Appl. Math., Springer.CrossRefGoogle Scholar
Kloeden, P. E. and Platen, E. (1995 a), Numerical methods for stochastic differential equations, in Nonlinear Dynamics and Stochastic Mechanics, CRC Math. Model. Series, CRC Press, Boca Raton, pp. 437–461.Google Scholar
Kloeden, P. E. and Scott, W. D. (1993), ‘Construction of stochastic numerical schemes through Maple’, Maple Technical Newspaper 10, 60–65.Google Scholar
Kloeden, P. E., Platen, E. and Hofmann, N. (1992 a), Stochastic differential equations: Applications and numerical methods, in Proceedings of the 6th IAHR International Symposium on Stochastic Hydraulics, National Taiwan University, Taipeh, pp. 75–81.Google Scholar
Kloeden, P. E., Platen, E. and Hofmann, N. (1995), ‘Extrapolation methods for the weak approximation of Itô diffusions’, SIAM J. Numer. Anal. 32, 1519–1534.CrossRefGoogle Scholar
Kloeden, P. E., Platen, E. and Schurz, H. (1991), ‘The numerical solution of nonlinear stochastic dynamical systems: A brief introduction’, J. Bifur. Chaos 1, 277–286.CrossRefGoogle Scholar
Kloeden, P. E., Platen, E. and Schurz, H. (1992 b), ‘Effective simulation of optimal trajectories in stochastic control’, Optimization 1, 633–644.Google Scholar
Kloeden, P. E., Platen, E. and Schurz, H. (1993), Higher order approximate Markov chain filters, in Stochastic Processes: A Festschrift in Honour of Gopinath Kallianpur (Cambanis, S. et al. , eds), Springer, pp. 181–190.CrossRefGoogle Scholar
Kloeden, P. E., Platen, E. and Schurz, H. (1994/1997), Numerical Solution of SDEs Through Computer Experiments, Universitext, Springer.CrossRefGoogle Scholar
Kloeden, P. E., Platen, E. and Wright, I. (1992c), ‘The approximation of multiple stochastic integrals’, Stoch. Anal. Appl. 10, 431–441.CrossRefGoogle Scholar
Kohatsu-Higa, A. (1997), ‘High order Itô–Taylor approximations to heat kernels’, J. Math. Kyoto Univ. 37, 129–150.Google Scholar
Kohatsu-Higa, A. and Ogawa, S. (1997), ‘Weak rate of convergence for an Euler scheme of nonlinear SDE's’, Monte Carlo Methods Appl. 3, 327–345.CrossRefGoogle Scholar
Kohatsu-Higa, A. and Protter, P. (1994), The Euler scheme for SDEs driven by semimartingales, in Stochastic Anal. on Infinite Dimensional Spaces (Kunita, H. and Kuo, H. H., eds), Pitman, pp. 141–151.Google Scholar
Kohler, W. E. and Boyce, W. E. (1974), ‘A numerical analysis of some first order stochastic initial value problems’, SIAM J. Appl. Math. 27, 167–179.CrossRefGoogle Scholar
Komori, Y. and Mitsui, T. (1995), ‘Stable ROW-type weak scheme for stochastic differential equations’, Monte Carlo Methods Appl. 1, 279–300.CrossRefGoogle Scholar
Komori, Y., Mitsui, T. and Sugiura, H. (1997), ‘Rooted tree analysis of the order conditions of ROW-type scheme for stochastic differential equations’, BIT 37, 43–66.CrossRefGoogle Scholar
Komori, Y., Saito, Y. and Mitsui, T. (1994), ‘Some issues in discrete approximate solution for stochastic differential equations’, Comput. Math. Appl. 28, 269–278.CrossRefGoogle Scholar
Kozlov, R. I. and Petryakov, M. G. (1986), ‘The construction of comparison systems for stochastic differential equations and numerical methods’, Nauka Sibirsk Otdel. Novosibirsk pp. 45–52. In Russian.Google Scholar
Kurtz, T. G. and Protter, P. (1991 a), ‘Weak limit theorems for stochastic integrals and stochastic differential equations’, Ann. Probab. 19, 1035–1070.CrossRefGoogle Scholar
Kurtz, T. G. and Protter, P. (1991 b), Wong–Zakai corrections, random evolutions and simulation schemes for SDE's, in Stochastic Analysis (Meyer-Wolf, E. M. E. and Schwartz, A., eds), Academic Press, pp. 331–346.CrossRefGoogle Scholar
Kushner, H. J. (1974), ‘On the weak convergence of interpolated Markov chains to a diffusion’, Ann. Probab. 2, 40–50.CrossRefGoogle Scholar
Kushner, H. J. and Dupuis, P. G. (1992), Numerical Methods for Stochastic Control Problems in Continuous Time, Vol. 24 of Applications of Mathematics, Springer, New York.CrossRefGoogle Scholar
Kuznetsov, D. F. (1998), Some Questions in the Theory of Numerical Solution of Itô Stochastic Differential Equations, Saint Petersburg, State Technical University Publisher. In Russian.Google Scholar
Law, A. M. and Kelton, W. D. (1991), Simulation Modeling and Analysis, 2nd edn, McGraw-Hill, New York.Google Scholar
LeGland, F. (1992), Splitting-up approximation for SPDEs and SDEs with application to nonlinear filtering, in Stochastic Partial Differential Equations and their Applications, Vol. 176 of Lecture Notes in Control and Inform. Sci., Springer, Berlin, pp. 177–187.CrossRefGoogle Scholar
Lépingle, D. (1993), ‘An Euler scheme for stochastic differential equations with reflecting boundary conditions’, Computes Rendus Acad. Sci. Paris, Séries I Math. 316, 601–605.Google Scholar
Lépingle, D. (1995), ‘Euler scheme for reflected stochastic differential equations’, Math. Comput. Simul. 38, 119–126.CrossRefGoogle Scholar
Lépingle, D. and Ribémont, B. (1991), ‘A multistep approximation scheme for the Langevin equation’, Stochastic Process. Appl. 37, 61–69.CrossRefGoogle Scholar
Li, C. W. and Liu, X. Q. (1997), ‘Algebraic structure of multiple stochastic integrals with respect to Brownian motions and Poisson processes’, Stochastics and Stochastics Reports 61, 107–120.CrossRefGoogle Scholar
Liske, H. (1982), ‘Distribution of a functional of a Wiener process’, Theory of Random Processes 10, 50–54. In Russian.Google Scholar
Liske, H. (1985), ‘Solution of an initial-boundary value problem for a stochastic equation of parabolic type by the semi-discretization method’, Theory of Random Processes 113, 51–56. In Russian.Google Scholar
Liske, H. and Platen, E. (1987), ‘Simulation studies on time discrete diffusion approximations’, Math. Comput. Simul. 29, 253–260.CrossRefGoogle Scholar
Liu, X. Q. and Li, C. W. (1997), ‘Discretization of stochastic differential equations by the product expansion for the Chen series’, Stochastics and Stochastics Reports 60, 23–40.CrossRefGoogle Scholar
Ma, J., Protter, P. and Yong, J. M. (1994), ‘Solving forward–backward stochastic differential equations explicitly: a four step scheme’, Probability Theory Related Fields 98, 339–359.CrossRefGoogle Scholar
Mackevicius, V. (1987), ‘Sp-stability of solutions of symmetric stochastic differential equations with discontinuous driving semimartingales’, Ann. Inst. H. Poincaré Probab. Statist. 23, 575–592.Google Scholar
Mackevicius, V. (1994), ‘Second order weak approximations for Stratonovich stochastic differential equations’, Lietuvos Matem. Rink. 34, 226–247. Translation in Lithuanian Math. Journal, 34, 183–200.Google Scholar
Mackevicius, V. (1996), Extrapolation of approximations of solutions of stochastic differential equations, in Probability Theory and Mathematical Statistics, World Scientific, River Edge, NJ, pp. 276–297.Google Scholar
Maghsoodi, Y. (1994), Mean-square efficient numerical solution of jump-diffusion stochastic differential equations, Preprint OR72, University of Southampton, UK.Google Scholar
Maghsoodi, Y. and Harris, C. J. (1987), ‘In-probability approximation and simulation of nonlinear jump-diffusion stochastic differential equations’, IMA J. Math. Control Inform. 4, 65–92.CrossRefGoogle Scholar
Makroglou, A. (1991), ‘Numerical treatment of stochastic Volterra integro-differential equations’, J. Comput. Appl. Math. II, 307–313.Google Scholar
Maltz, F. H. and Hitzl, D. L. (1979), ‘Variance reduction in Monte-Carlo computations using multi-dimensional Hermite polynomials’, J. Comput. Phys. 32, 345–376.CrossRefGoogle Scholar
Manella, R. and Palleschi, V. (1989), ‘Fast and precise algorithm for computer simulation of stochastic differential equations’, Phys. Rev. A 40, 3381–3386.CrossRefGoogle Scholar
Marcus, S. I. (1981), ‘Modeling and approximation of stochastic differential equations driven by semimartingales’, Stochastics 4, 223–245.CrossRefGoogle Scholar
Marsaglia, G. and Bray, T. A. (1964), ‘A convenient method for generating normal variables’, SIAM Review 6, 260–264.CrossRefGoogle Scholar
Maruyama, G. (1955), ‘Continuous Markov processes and stochastic equations’, Rend. Circolo Math. Palermo 4, 48–90.CrossRefGoogle Scholar
Mauthner, S. (1998), ‘Step size control in the numerical solution of stochastic differential equations’, J. Comput. Appl. Math. 100, 93–109.CrossRefGoogle Scholar
Merton, R. (1973), ‘The theory of rational option pricing’, Bell Journal of Economics and Management Science 4, 141–183.Google Scholar
Mikhailov, G. A. (1992), Optimization of Weighted Monte Carlo Methods. Series in Computational Physics, Springer.CrossRefGoogle Scholar
Mikulevicius, R. and Platen, E. (1988), ‘Time discrete Taylor approximations for Itô processes with jump component’, Mathematische Nachrichten 138, 93–104.CrossRefGoogle Scholar
Mikulevicius, R. and Platen, E. (1991), ‘Rate of convergence of the Euler approximation for diffusion processes’, Mathematische Nachrichten 151, 233–239.CrossRefGoogle Scholar
Milstein, G. N. (1974), ‘Approximate integration of stochastic differential equations’, Theory Probab. Appl. 19, 557–562.Google Scholar
Milstein, G. N. (1978), ‘A method of second order accuracy integration of stochastic differential equations’, Theory Probab. Appl. 23, 396–401.CrossRefGoogle Scholar
Milstein, G. N. (1985), ‘Weak approximation of solutions of systems of stochastic differential equations’, Theory Probab. Appl. 30, 750–766.CrossRefGoogle Scholar
Milstein, G. N. (1987), ‘A theorem on the order of convergence of mean-square approximations of solutions of systems of stochastic differential equations’, Teor. Veroyatnost. i Primenen 32, 809–811. In Russian.Google Scholar
Milstein, G. N. (1988 a), Numerical Integration of Stochastic Differential Equations, Urals Univ. Press, Sverdlovsk. In Russian.Google Scholar
Milstein, G. N. (1988 b), ‘A theorem of the order of convergence of mean square approximations of systems of stochastic differential equations’, Theory Probab. Appl. 32, 738–741.CrossRefGoogle Scholar
Milstein, G. N. (1995 a), Numerical Integration of Stochastic Differential Equations, Mathematics and its Applications, Kluwer, Dordrecht/Boston/London.CrossRefGoogle Scholar
Milstein, G. N. (1995 b), ‘The solving of boundary value problems by numerical integration of stochastic equations’, Math. Comput. Simul. 38, 77–85.Google Scholar
Milstein, G. N. (1995 c), ‘Solving the first boundary value problem of parabolic type by numerical integration of stochastic differential equations’, Theory Probab. Appl. 40, 657–665.Google Scholar
Milstein, G. N. (1996), ‘Application of numerical integration of stochastic equations for solving boundary value problems with Neumann boundary conditions’, Theory Probab. Appl. 41, 210–218.Google Scholar
Milstein, G. N. (1997), ‘Weak approximation of a diffusion process in a bounded domain’, Stochastics and Stochastics Reports 62, 147–200.CrossRefGoogle Scholar
Milstein, G. N. and Platen, E. (1994), The integration of stiff stochastic differential equations with stable second moments, Technical report SRR 014–94, Australian National University Statistics Report Series.Google Scholar
Milstein, G. N. and Tretjakov, M. V. (1994), ‘Numerical solution of differential equations with colored noise’, J. Statist. Phys. 77, 691–715.CrossRefGoogle Scholar
Milstein, G. N. and Tretjakov, M. V. (1997), ‘Numerical methods in the weak sense for stochastic differential equations with small noise’, SIAM J. Numer. Anal. 34, 2142–2167.CrossRefGoogle Scholar
Milstein, G. N., Platen, E. and Schurz, H. (1998), ‘Balanced implicit methods for stiff stochastic systems’, SIAM J. Numer. Anal. 35, 1010–1019.CrossRefGoogle Scholar
Morgan, B. J. (1984), Elements of Simulation, Chapman & Hall, London.CrossRefGoogle Scholar
Mori, M. (1998), ‘Low discrepancy sequences generated by piecewise linear maps’, Monte Carlo Methods Appl. 4, 141–162.CrossRefGoogle Scholar
Müller-Gronbach, T. (1996), ‘Optimal design for approximating the path of a stochastic process’, J. Statist. Planning Inf. 49, 371–385.CrossRefGoogle Scholar
Nakazawa, H. (1990), ‘Numerical procedures for sample structures on stochastic differential equations’, J. Math. Phys. 31, 1978–1990.CrossRefGoogle Scholar
Newton, N. J. (1986 a), ‘An asymptotic efficient difference formula for solving stochastic differential equations’, Stochastics 19, 175–206.CrossRefGoogle Scholar
Newton, N. J. (1986 b), Asymptotically optimal discrete approximations for stochastic differential equations, in Theory and Applications of Nonlinear Control Systems, North-Holland, pp. 555–567.Google Scholar
Newton, N. J. (1990), ‘An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times’, Stochastics 29, 227–258.Google Scholar
Newton, N. J. (1991), ‘Asymptotically efficient Runge–Kutta methods for a class of Itô and Stratonovich equations’, SIAM J. Appl. Math. 51, 542–567.CrossRefGoogle Scholar
Newton, N. J. (1994), ‘Variance reduction for simulated diffusions’, SIAM J. Appl. Math. 54, 1780–1805.CrossRefGoogle Scholar
Newton, N. J. (1996), ‘Numerical methods for stochastic differential equations’, Z. Angew. Math. Mech. 76, 211–214. Suppl. 3, I–XVI.Google Scholar
Newton, N. J. (1997), Continuous-time Monte Carlo methods and variance reduction, in Numerical Methods in Finance, Newton Institute, Cambridge University Press, Cambridge, pp. 22–42.CrossRefGoogle Scholar
Niederreiter, H. (1988), ‘Remarks on nonlinear pseudo random numbers’, Metrika 35, 321–328.CrossRefGoogle Scholar
Niederreiter, H. (1992), Random Number Generation and Quasi-Monte-Carlo Methods, SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Niederreiter, H. and Shine, P. J. (1995), Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Vol. 106 of Lecture Notes in Statistics, Springer.CrossRefGoogle Scholar
Nikitin, N. N. and Razevig, V. D. (1978), ‘Methods of numerical modelling of stochastic differential equations and estimates of their error’, Zh. Vychisl. Mat. Mat. Fiz 18, 106–117. In Russian.Google Scholar
Ogawa, S. (1992), ‘Monte Carlo simulation of nonlinear diffusion processes’, Japan J. Industrial and Appl. Math. 9, 25–33.CrossRefGoogle Scholar
Ogawa, S. (1994), ‘Monte Carlo simulation of nonlinear diffusion processes II’, Japan J. Industrial and Appl. Math. 2, 31–45.CrossRefGoogle Scholar
Ogawa, S. (1995), ‘Some problems in the simulation of nonlinear diffusion processes’, Math. Comput. Simul. 38, 217–223.CrossRefGoogle Scholar
Ogorodnikov, V. A. and Prigarin, S. M. (1996), Numerical Modelling of Random Processes and Fields: Algorithms and Applications, VSP, Utrecht.CrossRefGoogle Scholar
Oksendahl, B. (1985), Stochastic Differential Equations, Springer.CrossRefGoogle Scholar
Pardoux, E. and Talay, D. (1985), ‘Discretization and simulation of stochastic differential equations’, Acta Appl. Math. 3, 23–47.CrossRefGoogle Scholar
Paskov, S. and Traub, J. (1995), ‘Faster valuation of financial derivatives’, J. Portfolio Manag., pp. 113–120.CrossRefGoogle Scholar
Petersen, W. P. (1987), Numerical simulation of Itô stochastic differential equations on supercomputers, in Random Media, Vol. 7 of IMA Vol. Math. Appl., Springer, pp. 215–228.CrossRefGoogle Scholar
Petersen, W. P. (1988), ‘Some vectorized random number generators for uniform, normal and Poisson distributions for CRAY X-MP’, J. Supercomputing 1, 318–335.CrossRefGoogle Scholar
Petersen, W. P. (1994 a), ‘Lagged Fibonacci series random number generators for the NEC SX-3’, Intern. J. High Speed Computing 6, 387–398.CrossRefGoogle Scholar
Petersen, W. P. (1994 b), ‘Some experiments on numerical simulations of stochastic differential equations and a new algorithm’, J. Comput. Phys. 113, 75–81.CrossRefGoogle Scholar
Petersen, W. P. (1998), ‘A general implicit splitting for stabilizing numerical simulations of Itô stochastic differential equations’, SIAM J. Numer. Anal. 35, 1439–1451.CrossRefGoogle Scholar
Petterson, R. (1995), ‘Approximations for stochastic differential equations with reflecting convex boundaries’, Stochastic Process. Appl. 59, 295–308.CrossRefGoogle Scholar
Platen, E. (1980 a), Approximation of Itô integral equations, in Stochastic differential systems, Vol. 25 of Lecture Notes in Control and Inform. Sci., Springer, pp. 172–176.Google Scholar
Platen, E. (1980 b), ‘Weak convergence of approximations of Itô integral equations’, Z. Angew. Math. Mech. 60, 609–614.CrossRefGoogle Scholar
Platen, E. (1981), ‘An approximation method for a class of Itô processes’, Lietuvos Matem. Rink. 21, 121–133.Google Scholar
Platen, E. (1982 a), ‘An approximation method for a class of Itô processes with jump component’, Lietuvos Matem. Rink. 22, 124–136.Google Scholar
Platen, E. (1982 b), ‘A generalized Taylor formula for solutions of stochastic differential equations’, SANKHYA A 44, 163–172.Google Scholar
Platen, E. (1983), ‘Approximation of first exit times of diffusions and approximate solution of parabolic equations’, Mathematische Nachrichten 111, 127–146.CrossRefGoogle Scholar
Platen, E. (1984), Zur zeitdiskreten Approximation von Itoprozessen, Diss. B., IMath, Akad. der Wiss. der DDR, Berlin.Google Scholar
Platen, E. (1985), On first exit times of diffusions, in Stochastic differential systems, Vol. 69 of Lecture Notes in Control and Inform. Sci., Springer, pp. 192–195.Google Scholar
Platen, E. (1987), Derivative free numerical methods for stochastic differential equations, Vol. 96 of Lecture Notes in Control and Inform. Sci., Springer, pp. 187–193.Google Scholar
Platen, E. (1992), ‘Higher-order weak approximation of Itô diffusions by Markov chains’, Probability in the Engineering and Information Sciences 6, 391–408.CrossRefGoogle Scholar
Platen, E. (1995), ‘On weak implicit and predictor-corrector methods’, Math. Comput. Simul. 38, 69–76.CrossRefGoogle Scholar
Platen, E. and Rebolledo, R. (1985), ‘Weak convergence of semimartingales and discretization methods’, Stochastic Process. Appl. 20, 41–58.CrossRefGoogle Scholar
Platen, E. and Wagner, W. (1982), ‘On a Taylor formula for a class of Itô processes’, Probability and Math. Statistics 3, 37–51.Google Scholar
Protter, P. (1985), ‘Approximations of solutions of stochastic differential equations driven by semimartingales’, Ann. Probab. 13, 716–743.Google Scholar
Protter, P. (1990), Stochastic Integration and Differential Equations, Springer.CrossRefGoogle Scholar
Protter, P. and Talay, D. (1997), ‘The Euler scheme for Levy driven stochastic differential equations’, Ann. Probab. 25, 393–423.CrossRefGoogle Scholar
Radovic, I., Sobol, I. M. and Tichy, R. F. (1996), ‘Quasi-Monte Carlo methods for numerical integration: Comparison of different low discrepancy sequences’, Monte Carlo Methods Appl. 2, 1–14.CrossRefGoogle Scholar
Rao, N. J., Borwankar, J. D. and Ramkrishna, D. (1974), ‘Numerical solution of Itô integral equations’, SIAM J. Control Optimiz. 12, 124–139.CrossRefGoogle Scholar
Razevig, V. D. (1980), ‘Digital modelling of multi-dimensional dynamics under random perturbations’, Autom. Remote Control 4, 177–186. In Russian.Google Scholar
Ripley, B. D. (1983 a), ‘Computer generation of random variables: A tutorial letter’, Inter. Statist. Rev. 45, 301–319.CrossRefGoogle Scholar
Ripley, B. D. (1983 b), Stochastic Simulation, Wiley, New York.Google Scholar
Römisch, W. and Wakolbinger, A. (1987), On the convergence rates of approximate solutions of stochastic equations, in Vol. 96 of Lecture Notes in Control and Inform. Sci., Springer, pp. 204–212.Google Scholar
Ross, S. M. (1991), A Course in Simulation, MacMillan, New York.Google Scholar
Rubinstein, R. Y. (1981), Simulation and the Monte Carlo Method, Wiley.CrossRefGoogle Scholar
Rümelin, W. (1982), ‘Numerical treatment of stochastic differential equations’, SIAM J. Numer. Anal. 19, 604–613.CrossRefGoogle Scholar
Ryashko, L. B. and Schurz, H. (1997), ‘Mean square stability analysis of some linear stochastic systems’, Dyn. Sys. Appl. 6, 165–189.Google Scholar
Sabelfeld, K. K. (1979), ‘On the approximate computation of Wiener integrals by Monte-Carlo method’, Zh. Vychisl. Mat. Mat. Fiz 19, 29–43. In Russian.Google Scholar
Saito, Y. and Mitsui, T. (1993 a), ‘Simulation of stochastic differential equations’, Ann. Inst. Statist. Math. 45, 419–432.CrossRefGoogle Scholar
Saito, Y. and Mitsui, T. (1993 b), ‘T-stability of numerical schemes for stochastic differential equations’, World Sci. Ser. Appl. Anal. 2, 333–344.Google Scholar
Saito, Y. and Mitsui, T. (1995), ‘S-series in the Wong-Zakai approximation for stochastic differential equations’, Vietnam J. Math. 23, 303–317.Google Scholar
Saito, Y. and Mitsui, T. (1996), ‘Stability analysis of numerical schemes for stochastic differential equations’, SIAM J. Numer. Anal. 33, 2254–2267.CrossRefGoogle Scholar
Schein, O. and Denk, G. (1998), ‘Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits’, J. Comput. Appl. Math. 100, 77–92.CrossRefGoogle Scholar
Schurz, H. (1996 a), ‘Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise’, Stoch. Anal. Appl. 14, 313–354.CrossRefGoogle Scholar
Schurz, H. (1996 b), ‘Numerical regularization for SDEs: Construction of nonnegative solutions’, Dyn. Sys. Appl. 5, 323–351.Google Scholar
Schurz, H. (1996 c), Stability, stationarity and boundedness of some implicit numerical methods for stochastic differential equations, PhD thesis, Humboldt University, Berlin.Google Scholar
Shimizu, A. and Kawachi, T. (1984), ‘Approximate solutions of stochastic differential equations’, Bull. Nagoya Inst. Tech. 36, 105–108.Google Scholar
Shinozuka, M. (1971), ‘Simulation of multivariate and multidimensional random differential processes’, J. Acoust. Soc. Amer. 49, 357–367.CrossRefGoogle Scholar
Shkurko, I. O. (1987), Numerical solution of linear systems of stochastic differential equations, in Numerical Methods for Statistics and Modeling, Novosibirsk, pp. 101–109. Collected Scientific Works, in Russian.Google Scholar
Sloan, I. H. and Wozniakowski, H. (1998), ‘When are quasi-Monte-Carlo algorithms efficient for high dimensional integrals?’, J. Complexity 14, 1–33.CrossRefGoogle Scholar
Slominski, L. (1994), ‘On approximation of solutions of multidimensional SDEs with reflecting boundary conditions’, Stochastic Process. Appl. 50, 197–219.CrossRefGoogle Scholar
Sobol, I. M. (1967), ‘The distribution of points in a cube and the approximate evaluation of integrals’, USSR Comput. Math. Math. Phys. 19, 86–112.CrossRefGoogle Scholar
Steele, J. M. and Stine, R. A. (1993), Mathematica and diffusions, in Economic and Financial Modeling with Mathematica, TELOS, Santa Clara, CA, pp. 192–213.CrossRefGoogle Scholar
Stoer, J. and Bulirsch, R. (1993), Introduction to Numerical Analysis, 2nd edn, Springer. (1st edn (1980).)CrossRefGoogle Scholar
Sugita, H. (1995), ‘Pseudo-random number generator by means of irrational rotation’, Monte Carlo Methods Appl. 1, 35–57.CrossRefGoogle Scholar
Sun, M. and Glowinski, R. (1994), ‘Pathwise approximation and simulation for the Zakai filtering equation through operator splitting’, Calcolo 30, 219–239.CrossRefGoogle Scholar
Sussmann, H. J. (1988), Product expansions of exponential Lie series and the discretization of stochastic differential equations, in Stochastic Differential Systems, Stochastic Control Theory and Applications (Fleming, W. and Lions, P. I., eds), Vol. 10 of IMA Vol. Math. Appl., Springer, pp. 563–582.CrossRefGoogle Scholar
Talay, D. (1982 a), Analyse Numérique des Equations Différentielles Stochastiques, PhD thesis, Université de Provence, Centre Saint Charles. Thèse 3ème Cycle.Google Scholar
Talay, D. (1982 b), ‘Convergence pour chaque trajectoire d'un schéme d'approximation des EDS’, Computes Rendus Acad. Sci. Paris, Séries I Math 295, 249–252.Google Scholar
Talay, D. (1983 a), How to discretize stochastic differential equations, in Nonlinear filtering and stochastic control, Vol. 972 of Lecture Notes in Math., Springer, pp. 276–292.CrossRefGoogle Scholar
Talay, D. (1983 b), ‘Résolution trajectorielle et analyse numérique des équations différentielles stochastiques’, Stochastics 9, 275–306.CrossRefGoogle Scholar
Talay, D. (1984), Efficient numerical schemes for the approximation of expectations of functionals of the solution of an SDE and applications, in Filtering and Control of Random Processes, Vol. 61 of Lecture Notes in Control and Inform. Sci., Springer, pp. 294–313.CrossRefGoogle Scholar
Talay, D. (1986), ‘Discrétisation d'une équation différentielle stochastique et calcul approché d'espérances de fonctionelles de la solution’, Modél Math. et Anal. Numér. 20, 141–179.CrossRefGoogle Scholar
Talay, D. (1987), Classification of discretization of diffusions according to an ergodic criterion, in Stochastic Modelling and Filtering, Vol. 91 of Lecture Notes in Control and Inform. Sci., Springer, pp. 207–218.CrossRefGoogle Scholar
Talay, D. (1990), ‘Second order discretization schemes of stochastic differential systems for the computation of the invariant law’, Stochastics and Stochastics Reports 29, 13–36.CrossRefGoogle Scholar
Talay, D. (1991), ‘Approximation of upper Lyapunov exponents of bilinear stochastic differential equations’, SIAM J. Numer. Anal. 28, 1141–1164.CrossRefGoogle Scholar
Talay, D. (1995), Simulation of stochastic differential systems, in Probabilistic Methods in Applied Physics (Krée, P. and Wedig, W., eds), Vol. 451 of Lecture Notes in Physics., Springer, Chapter 3, pp. 54–96.CrossRefGoogle Scholar
Talay, D. and Tubaro, L. (1990), ‘Expansion of the global error for numerical schemes solving stochastic differential equations’, Stoch. Anal. Appl. 8, 483–509.CrossRefGoogle Scholar
Tetzlaff, U. and Zschiesche, H.-U. (1984), ‘Näherungslösungen für Itô-Differential-gleichungen mittels Taylorentwicklungen für Halbgruppen von Operatoren’, Wiss. Z. Techn. Hochschule Leuna-Merseburg 2, 332–339.Google Scholar
Tezuka, S. (1993), ‘Polynomial arithmetic analogue of Halton sequences’, ACM Trans. Model. Comput. Simul. 3, 99–107.CrossRefGoogle Scholar
Tezuka, S. and Tokuyama, T. (1994), ‘A note on polynomial arithmetic analogue of Halton sequences’, ACM Trans. Model. Computer Simul. 4, 279–284.CrossRefGoogle Scholar
Törok, C. (1994), ‘Numerical solution of linear stochastic differential equations’, Comput. Math. Appl. 27, 1–10.CrossRefGoogle Scholar
Traub, J. F., Wasilkowski, G. W. and Wozniakowski, H. (1988), Information-Based Complexity, Academic Press, New York.Google Scholar
Tudor, C. (1989), ‘Approximation of delay stochastic equations with constant retardation by usual Itô equations’, Rev. Roumaine Math. Pures Appl. 34, 55–64.Google Scholar
Tudor, C. and Tudor, M. (1983), ‘On approximation in quadratic mean for the solutions of two parameter stochastic differential equations in Hilbert spaces’, An. Univ. Bucuresti Mat. 32, 73–88.Google Scholar
Tudor, C. and Tudor, M. (1987), ‘On approximation of solutions for stochastic delay equations’, Stud. Cerc. Mat. 39, 265–274.Google Scholar
Tudor, C. and Tudor, M. (1995), ‘Approximation schemes for Itô-Volterra stochastic equations’, Bol. Soc. Mat. Mexicana (3) 1, 73–85.Google Scholar
Tudor, C. and Tudor, M. (1997), ‘Approximate solutions for multiple stochastic equations with respect to semimartingales’, Z. Anal. Anwendungen 16, 761–768.CrossRefGoogle Scholar
Tudor, M. (1992), ‘Approximation schemes for two-parameter stochastic equations’, Probability and Math. Statistics 13, 177–189.Google Scholar
Tuffin, B. (1996), ‘On the use of low discrepancy sequences in Monte Carlo methods’, Monte Carlo Methods Appl. 2, 295–320.CrossRefGoogle Scholar
Tuffin, B. (1997), ‘Comments on “On the use of low discrepancy sequences in Monte Carlo methods”’, Monte Carlo Methods Appl. 4, 87–90.Google Scholar
Unny, T. E. (1984), ‘Numerical integration of stochastic differential equations in catchment modelling’, Water Res. 20, 360–368.CrossRefGoogle Scholar
Valkeila, E. (1991), ‘Computer algebra and stochastic analysis’, CWI 4, 229–238.Google Scholar
Ventzel, A. D., Gladyshev, S. A. and Milstein, G. N. (1985), ‘Piecewise constant approximation for the Monte-Carlo calculation of Wiener integrals’, Theory Probab. Appl. 24, 745–752.Google Scholar
Wagner, W. (1987), ‘Unbiased Monte-Carlo evaluation of certain functional integrals’, J. Comput. Phys. 71, 21–33.CrossRefGoogle Scholar
Wagner, W. (1988 a), ‘Monte-Carlo evaluation of functionals of solutions of stochastic differential equations. Variance reduction and numerical examples’, Stoch. Anal. Appl. 6, 447–468.CrossRefGoogle Scholar
Wagner, W. (1988 b), ‘Unbiased multi-step estimators for the Monte-Carlo evaluation of certain functionals’, J. Comput. Phys. 79, 336–352.CrossRefGoogle Scholar
Wagner, W. (1989 a), Stochastische numerische Verfahren zur Berechnung von Funktionalintegralen, Habilitation, Report 02/89, IMATH, Berlin.Google Scholar
Wagner, W. (1989 b), ‘Unbiased Monte-Carlo estimators for functionals of weak solutions of stochastic differential equations’, Stochastics and Stochastics Reports 28, 1–20.CrossRefGoogle Scholar
Wagner, W. and Platen, E. (1978), Approximation of Itô integral equations, Preprint ZIMM, Akad. Wissenschaften, DDR, Berlin.Google Scholar
Werner, M. J. and Drummond, P. D. (1997), ‘Robust algorithms for solving stochastic partial differential equations’, J. Comput. Phys. 132, 312–326.CrossRefGoogle Scholar
Wiener, N. (1923), ‘Differential space’, J. Math. Phys. 2, 131–174.CrossRefGoogle Scholar
Wong, E. and Zakai, M. (1965), ‘On the convergence of ordinary integrals to stochastic integrals’, Ann. Math. Statist. 36, 1560–1564.CrossRefGoogle Scholar
Wozniakowski, H. (1991), ‘Average case complexity of multivariate integration’, Bull. Amer. Math. Soc. 24, 185–194.CrossRefGoogle Scholar
Wright, D. J. (1974), ‘The digital simulation of stochastic differential equations’, IEEE Trans. Automat. Control 19, 75–76.CrossRefGoogle Scholar
Wright, D. J. (1980), ‘Digital simulation of Poisson stochastic differential equations’, Intern. J. Systems. Sci, 11, 781–785.CrossRefGoogle Scholar
Xu, K. (1995), ‘Stochastic pitchforkbifurcation: numerical simulations and symbolic calculations using Maple’, Math. Comput. Simul. 38, 199.CrossRefGoogle Scholar
Yakowitz, S. J. (1977), Computational Probability and Simulation, Addison Wesley, Reading, MA.Google Scholar
Yamada, T. (1976), ‘Sur l'approximation des solutions d'équations différentielles stochastiques’, Z. Wahrsch. Verw. Gebiete 36, 153–164.CrossRefGoogle Scholar
Yannios, N. and Kloeden, P. E. (1996), Time-discretization solution of stochastic differential equations, in Proc. CTAC 95 (May, R. L. and Easton, A. K., eds), Computational Techniques and Applications: CTAC95, World Scientific, pp. 823–830.Google Scholar
Yen, Y. Y. (1988), ‘A stochastic Taylor formula for functionals of two-parameter semimartingales’, Acta Vietnamica 13, 45–54.Google Scholar