An introduction to numerical methods for stochastic differential equations

Acta Numerica

Volumn 8, Issue , 1999, Pages 197-246

  Platen, Eckhard ^a  

^a UNIVERSITY OF TECHNOLOGY SYDNEY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85011454437     PISSN: 09624929     EISSN: 14740508     Source Type: Journal    
DOI: 10.1017/S0962492900002920     Document Type: Article

References (349)

1. 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.
2. Allain M. F. (1974), Sur quelques types d'approximation des solutions d'équations différentielles stochastiques, PhD thesis, Univ. Rennes.
3. 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.
4. Anderson S. L. (1990), ‘Random number generators on vector supercomputers and other advanced structures’, SIAM Review 32, 221–251.
5. 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.
6. Antipov M. V. (1996), ‘Sequences of numbers for Monte Carlo methods’, Monte Carlo Methods Appl. 2, 219–235.
7. Arnold L. (1974), Stochastic Differential Equations, Wiley, New York.
8. Arnold L. and Kloeden P. E. (1996), ‘Discretization of a random dynamical system near a hyperbolic point’, Mathematische Nachrichten 181, 43–72.
9. 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.
10. 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.
11. 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.
12. 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.
13. Artemiev S. S. and Averina A. T. (1997), Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations, VSP, Utrecht.
14. 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.
15. Asmussen S., Glynn P. and Pitman J. (1995), ‘Discretization error in simulation of one-dimensional reflecting Brownian motion’, Ann. Appl. Probab. 5, 875–896.
16. 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.
17. 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.
18. 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.
19. 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.
20. Bachelier L. (1900), ‘Théorie de la spéculation’, Annales de l'Ecole Normale Supérieure, Series 3 17, 21–86.
21. p-convergence’, Stochastics and Stochastics Reports 28, 209–246.
22. Bally V. (1989 b), ‘Approximation for the solution of stochastic differential equations. II: Strong-convergence’, Stochastics and Stochastics Reports 28, 357–385.
23. Bally V. (1990), ‘Approximation for the solutions of stochastic differential equations. III: Jointly weak convergence’, Stochastics and Stochastics Reports 30, 171–191.
24. Bally V. and Talay D. (1995), ‘The Euler scheme for stochastic differential equations: Error analysis with Malliavin calculus’, Math. Comput. Simul. 38, 35–41.
25. 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.
26. 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.
27. Barraquand J. (1995), ‘Monte Carlo integration, quadratic resampling, and asset pricing’, Math. Comput. Simul. 38, 173–182.
28. BenArous G. (1989), ‘Flots et series de Taylor stochastiques’, Probability Theory Related Fields 81, 29–77.
29. 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.
30. 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.
31. Björck A. and Dahlquist G. (1974), Numerical Methods. Series in Automatic Computation, Prentice-Hall, New York.
32. Black F. and Scholes M. (1973), ‘The pricing of options and corporate liabilities’, J. Political Economy 81, 637–659.
33. Bouleau N. (1990), ‘On effective computation of expectations in large or infinite dimension: Random numbers and simulation’, J. Comput. Appl. Math. 31, 23–34.
34. Bouleau N. and Lépingle D. (1993), Numerical Methods for Stochastic Processes, Wiley, New York.
35. Box G. and Muller M. (1958), ‘A note on the generation of random normal variables’, Ann. Math. Statist. 29, 610–611.
36. Boyce W. E. (1978), ‘Approximate solution of random ordinary differential equations’, Adv. Appl. Probab. 10, 172–184.
37. Boyle P. P. (1977), ‘A Monte Carlo approach’, J. Financial Economics 4, 323–338.
38. Bratley P., Fox B. L. and Schrage L. (1987), A Guide to Simulation, 2nd edn, Springer, New York.
39. Brent R. P. (1974), ‘A Gaussian pseudo number generator’, Commun. Assoc. Comput. Mach. 17, 704–706.
40. Burrage K. (1995), Parallel and Sequential Methods for Ordinary Differential Equations, Clarendon Press, Oxford University Press.
41. 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.
42. 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.
43. Burrage K. and Platen E. (1994), ‘Runge–Kutta methods for stochastic differential equations’, Ann. Numer. Math. 1, 63–78.
44. 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.
45. Burrage P. M. (1998), Runge–Kutta methods for stochastic differential equations, PhD thesis, University of Queensland, Brisbane, Australia.
46. Butcher J. C. (1987), The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods, Wiley, Chichester.
47. 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.
48. 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.
49. 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.
50. 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.
51. 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.
52. 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.
53. Chang C. C. (1987), ‘Numerical solution of stochastic differential equations with constant diffusion coefficients’, Math. Comput. 49, 523–542.
54. 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.
55. 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.
56. 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.
57. 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.
58. 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.
59. 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.
60. 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.
61. Cyganowski S. O. (1996), ‘Solving stochastic differential equations with Maple’, Maple Tech. 3, 38.
62. 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.
63. Denk G. and Schäffer S. (1997), ‘Adam's methods for the efficient solution of stochastic differential equations with additive noise’, Computing 59, 153–161.
64. Douglas J., Ma J. and Protter P. (1996), ‘Numerical methods for forward-backward stochastic differential equations’, Ann. Appl. Probab. 6, 940–968.
65. 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.
66. 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.
67. Drummond P. D. and Mortimer I. K. (1991), ‘Computer simulation of multiplicative stochastic differential equations’, J. Comput. Phys. 93, 144–170.
68. 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.
69. Eichenauer J. and Lehn J. (1986), ‘A non-linear congruential pseudo random number generator’, Statist. Paper 27, 315–326.
70. Einstein A. (1906), ‘Zur Theorie der Brownschen Bewegung’, Ann. Phys. IV 19, 371.
71. Elliott R. J. (1982), Stochastic Calculus and Applications, Springer.
72. Elliott R. J. and Glowinski R. (1989), ‘Approximations to solutions of the Zakai filtering equation’, Stoch. Anal. Appl. 7, 145–168.
73. 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.
74. 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.
75. Ermakov S. M. and Mikhailov (1982), Statistical Modeling, 2nd edn, Nauka, Moscow.
76. Fahrmeier L. (1974), ‘Schwache Konvergenz gegen Diffusionsprozesse’, Z. Angew. Math. Mech. 54, 245.
77. Fahrmeier L. (1976), ‘Approximation von stochastischen Differenzialgleichungen auf Digital- und Hybridrechnern’, Computing 16, 359–371.
78. Feng J. F. (1990), ‘Numerical solution of stochastic differential equations’, Chinese J. Numer. Appl. 12, 28–41.
79. 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.
80. 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.
81. Fishman G. S. (1992), Monte Carlo: Concepts, Algorithms and Applications. Series in Operations Research, Springer.
82. Fournie E., Lebuchoux J. and Touzi N. (1997), ‘Small noise expansion and importance sampling’, Asympt. Anal. 14, 331–376.
83. Fox R. F. (1991), ‘Second-order algorithm for the numerical integration of colored-noise problems’, Phys. Rev. A 43, 2649–2654.
84. Franklin J. N. (1965), ‘Difference methods for stochastic ordinary differential equations’, Math. Comput. 19, 552–561.
85. Gaines J. G. (1994), ‘The algebra of iterated stochastic integrals’, Stochastics and Stochastics Reports 49, 169–179.
86. Gaines J. G. (1995 a), ‘A basis for iterated stochastic integrals’, Math. Comput. Simul. 38, 7–11.
87. Gaines J. G. (1995 b), Numerical experiments with S(P)DE's, in Proceedings of the ICMS Conference March 1994, Cambridge University Press.
88. Gaines J. G. and Lyons T. J. (1994), ‘Random generation of stochastic area integrals’, SIAM J. Appl. Math. 54, 1132–1146.
89. 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.
90. Gard T. C. (1988), Introduction to Stochastic Differential Equations, Marcel Dekker, New York.
91. Gear C. W. (1971), Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ.
92. Gelbrich M. (1995), ‘Simultaneous time and chance discretization for stochastic differential equations’, J. Comput. Appl. Math. 58, 255–289.
93. p 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.
94. Gentle J. E. (1998), Random Number Generation and Monte Carlo Methods. Series in Statistics and Computing, Springer.
95. Gerardi A., Marchetti F. and Rosa A. M. (1984), ‘Simulation of diffusions with boundary conditions’, Systems Control Lett. 4, 253.
96. Gikhman I. I. and Skorokhod A. V. (1979), The Theory of Stochastic Processes, Vol. I–III, Springer.
97. 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.
98. 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.
99. Glynn P. W. and Iglehart O. L. (1989), ‘Importance sampling for stochastic simulations’, Management Science 35, 1367–1392.
100. Golec J. (1995), ‘Stochastic averaging principle for systems with pathwise uniqueness’, Stoch. Anal. Appl. 13, 307–322.
101. Golec J. (1997), ‘Averaging Euler-type difference schemes’, Stoch. Anal. Appl. 15, 751–758.
102. Golec J. and Ladde G. S. (1989), ‘Euler-type approximation for systems of stochastic differential equations’, J. Appl. Math. Simul. 2, 239–249.
103. Golec J. and Ladde G. S. (1990), ‘Averaging principle and systems of singularly perturbed stochastic differential equations’, J. Math. Phys. 31, 1116–1123.
104. 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.
105. 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.
106. Grecksch W. and Kloeden P. E. (1996), ‘Time-discretised Galerkin approximations of parabolic stochastic PDEs’, Bull. Austral. Math. Soc. 54, 79–85.
107. Grecksch W. and Wadewitz A. (1996), ‘Approximation of solutions of stochastic differential equations by discontinuous Galerkin methods’, J. Anal. Appl. 15, 901–916.
108. Greenside H. S. and Helfand E. (1981), ‘Numerical integration of stochastic differential equations’, Bell Syst. Techn. J. 60, 1927–1940.
109. Greiner A., Strittmatter W. and Honerkamp J. (1987), ‘Numerical integration of stochastic differential equations’, J. Statist. Phys. 51, 95–108.
110. 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.
111. Grorud A. and Talay D. (1996), ‘Approximation of Lyapunov exponents of nonlinear stochastic differential equations’, SIAM J. Appl. Math. 56, 627–650.
112. Guo S. J. (1982), ‘On the mollifier approximation for solutions of stochastic differential equations’, J. Math. Kyoto Univ. 22, 243–254.
113. Guo S. J. (1984), ‘Approximation theorems based on random partitions for stochastic differential equations and applications’, Chinese Ann. Math. 5, 169–183.
114. Gyöngy I. (1991), ‘On approximation of Itô stochastic equations’, Math. SSR Sbornik 70, 165–173.
115. Gyöngy I. and Nuarlart D. (1997), ‘Implicit scheme for stochastic partial differential equations driven by space-time white noise’, Potential Analysis 7, 725–757.
116. Hairer E. and Wanner G. (1991), Solving ordinary differential equations II: Stiff and differential algebraic systems, Springer.
117. Hairer E., Nørsett S. P. and Wanner G. (1987), Solving ordinary differential equations I: Nonstiff problems, Springer.
118. Halton J. H. (1960), ‘On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals’, Numer. Math. 2, 84–90.
119. Hammersley J. M. and Handscomb D. C. (1964), Monte Carlo Methods, Methuen, London.
120. 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.
121. 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.
122. 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.
123. 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.
124. Heath D. and Platen E. (1996), ‘Valuation of FX barrier options under stochastic volatility’, Financial Engineering and the Japanese Markets 3, 195–215.
125. Helfand E. (1979), ‘Numerical integration of stochastic differential equations’, Bell Syst. Techn. J. 58, 2289–2299.
126. Hernandez D. B. and Spigler R. (1992), ‘ A-stability of implicit Runge–Kutta methods for systems with additive noise’, BIT 32, 620–633.
127. Hernandez D. B. and Spigler R. (1993), ‘Convergence and stability of implicit Runge–Kutta methods for systems with multiplicative noise’, BIT 33, 654–669.
128. 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.
129. Hofmann N. (1994), Beiträge zur schwachen Approximation stochastischer Differentialgleichungen, PhD thesis, Dissertation Humboldt Universität Berlin.
130. Hofmann N. (1995), ‘Stability of weak numerical schemes for stochastic differential equations’, Math. Comput. Simul. 38, 63–68.
131. Hofmann N. and Mathé P. (1997), ‘On quasi-Monte Carlo simulation of stochastic differential equations’, Math. Comput. 66, 573–589.
132. Hofmann N. and Platen E. (1994), ‘Stability of weak numerical schemes for stochastic differential equations’, Comput. Math. Appl. 28, 45–57.
133. Hofmann N. and Platen E. (1996), ‘Stability of superimplicit numerical methods for stochastic differential equations’, Fields Institute Communications 9, 93–104.
134. 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.
135. Hofmann N., Platen E. and Schweizer M. (1992), ‘Option pricing under incompleteness and stochastic volatility’, Mathematical Finance 2, 153–187.
136. 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.
137. 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.
138. Hu Y. Z. and Meyer P. A. (1993), ‘On the approximation of multiple Stratonovich integrals’, in Stochastic Processes, Springer, pp. 141–147.
139. 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.
140. Hull J. and White A. (1988), ‘The use of control variate techniques in option pricing’, J. Financial and Quantitative Analysis 23, 237–251.
141. Ikeda N. and Watanabe S. (1989), Stochastic Differential Equations and Diffusion Processes, 2nd edn, North-Holland, Amsterdam. (1st edn (1981).)
142. Itô K. (1944), ‘Stochastic integral’, Proc. Imp. Acad. Tokyo 20, 519–524.
143. Jacod J. and Protter P. (1998), ‘Asymptotic error distribution for the Euler method for stochastic differential equations’, Ann. Probab. 26, 267–307.
144. Jacod J. and Shiryaev A. N. (1987), Limit Theorems for Stochastic Processes, Springer.
145. 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.
146. 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.
147. Janicki A., Michna Z. and Weron A. (1996), ‘Approximation of stochastic differential equations driven by α-stable Lévy motion’, Applicationes Mathematicae 24, 149–168.
148. Janssen R. (1984 a), ‘Difference-methods for stochastic differential equations with discontinuous coefficients’, Stochastics 13, 199–212.
149. Janssen R. (1984 b), ‘Discretization of the Wiener process in difference methods for stochastic differential equations’, Stochastic Process. Appl. 18, 361–369.
150. Joy C., Boyle P. P. and Tan K. S. (1996), ‘Quasi Monte Carlo methods in numerical finance’, Management Science 42, 926–938.
151. Kalos M. H. and Whitlock P. A. (1986), Monte Carlo Methods, Wiley-Interscience, New York.
152. Kanagawa S. (1988), ‘The rate of convergence for Maruyama's approximate solutions of stochastic differential equations’, Yokohama Math. J. 36, 79–85.
153. Kanagawa S. (1989), ‘The rate of convergence for approximate solutions of stochastic differential equations’, Tokyo J. Math. 12, 33–48.
154. Kanagawa S. (1995), ‘Error estimation for the Euler–Maruyama approximate solutions of stochastic differential equations’, Monte Carlo Methods Appl. 1, 165–171.
155. 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.
156. 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.
157. 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.
158. 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.
159. Karatzas I. and Shreve S. E. (1988), Brownian Motion and Stochastic Calculus, Springer.
160. Kendall W. S. (1993), Doing stochastic calculus with Mathematica, in Economic and financial modeling with Mathematica, TELOS, Santa Clara, CA, pp. 214–238.
161. Klauder J. R. and Petersen W. P. (1985), ‘Numerical integration of multiplicative-noise stochastic differential equations’, SIAM J. Numer. Anal. 6, 1153–1166.
162. Kloeden P. E. and Pearson R. A. (1977), ‘The numerical solution of stochastic differential equations’, J. Austral. Math. Soc. Ser. B 20, 8–12.
163. Kloeden P. E. and Platen E. (1989), ‘A survey of numerical methods for stochastic differential equations’, J. Stochastic Hydrology and Hydraulics 3, 155–178.
164. Kloeden P. E. and Platen E. (1991 a), ‘Relations between multiple Itô and Stratonovich integrals’, Stoch. Anal. Appl. IX, 86–96.
165. Kloeden P. E. and Platen E. (1991 b), ‘Stratonovich and Itô stochastic Taylor expansions’, Mathematische Nachrichten 151, 33–50.
166. Kloeden P. E. and Platen E. (1992), ‘Higher order implicit strong numerical schemes for stochastic differential equations’, J. Statist. Phys. 66, 283–314.
167. Kloeden P. E. and Platen E. (1992/1995 b), Numerical Solution of Stochastic Differential Equations, Vol. 23 of Appl. Math., Springer.
168. 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.
169. Kloeden P. E. and Scott W. D. (1993), ‘Construction of stochastic numerical schemes through Maple’, Maple Technical Newspaper 10, 60–65.
170. 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.
171. 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.
172. 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.
173. Kloeden P. E., Platen E. and Schurz H. (1992 b), ‘Effective simulation of optimal trajectories in stochastic control’, Optimization 1, 633–644.
174. 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.
175. Kloeden P. E., Platen E. and Schurz H. (1994/1997), Numerical Solution of SDEs Through Computer Experiments, Universitext, Springer.
176. Kloeden P. E., Platen E. and Wright I. (1992c), ‘The approximation of multiple stochastic integrals’, Stoch. Anal. Appl. 10, 431–441.
177. Kohatsu-Higa A. (1997), ‘High order Itô–Taylor approximations to heat kernels’, J. Math. Kyoto Univ. 37, 129–150.
178. 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.
179. 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.
180. 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.
181. Komori Y. and Mitsui T. (1995), ‘Stable ROW-type weak scheme for stochastic differential equations’, Monte Carlo Methods Appl. 1, 279–300.
182. 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.
183. Komori Y., Saito Y. and Mitsui T. (1994), ‘Some issues in discrete approximate solution for stochastic differential equations’, Comput. Math. Appl. 28, 269–278.
184. 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.
185. Kurtz T. G. and Protter P. (1991 a), ‘Weak limit theorems for stochastic integrals and stochastic differential equations’, Ann. Probab. 19, 1035–1070.
186. 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.
187. Kushner H. J. (1974), ‘On the weak convergence of interpolated Markov chains to a diffusion’, Ann. Probab. 2, 40–50.
188. 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.
189. 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.
190. Law A. M. and Kelton W. D. (1991), Simulation Modeling and Analysis, 2nd edn, McGraw-Hill, New York.
191. 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.
192. 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.
193. Lépingle D. (1995), ‘Euler scheme for reflected stochastic differential equations’, Math. Comput. Simul. 38, 119–126.
194. Lépingle D. and Ribémont B. (1991), ‘A multistep approximation scheme for the Langevin equation’, Stochastic Process. Appl. 37, 61–69.
195. 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.
196. Liske H. (1982), ‘Distribution of a functional of a Wiener process’, Theory of Random Processes 10, 50–54. In Russian.
197. 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.
198. Liske H. and Platen E. (1987), ‘Simulation studies on time discrete diffusion approximations’, Math. Comput. Simul. 29, 253–260.
199. 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.
200. 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.
201. p-stability of solutions of symmetric stochastic differential equations with discontinuous driving semimartingales’, Ann. Inst. H. Poincaré Probab. Statist. 23, 575–592.
202. 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.
203. 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.
204. Maghsoodi Y. (1994), Mean-square efficient numerical solution of jump-diffusion stochastic differential equations, Preprint OR72, University of Southampton, UK.
205. 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.
206. Makroglou A. (1991), ‘Numerical treatment of stochastic Volterra integro-differential equations’, J. Comput. Appl. Math. II, 307–313.
207. 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.
208. Manella R. and Palleschi V. (1989), ‘Fast and precise algorithm for computer simulation of stochastic differential equations’, Phys. Rev. A 40, 3381–3386.
209. Marcus S. I. (1981), ‘Modeling and approximation of stochastic differential equations driven by semimartingales’, Stochastics 4, 223–245.
210. Marsaglia G. and Bray T. A. (1964), ‘A convenient method for generating normal variables’, SIAM Review 6, 260–264.
211. Maruyama G. (1955), ‘Continuous Markov processes and stochastic equations’, Rend. Circolo Math. Palermo 4, 48–90.
212. Mauthner S. (1998), ‘Step size control in the numerical solution of stochastic differential equations’, J. Comput. Appl. Math. 100, 93–109.
213. Merton R. (1973), ‘The theory of rational option pricing’, Bell Journal of Economics and Management Science 4, 141–183.
214. Mikhailov G. A. (1992), Optimization of Weighted Monte Carlo Methods. Series in Computational Physics, Springer.
215. Mikulevicius R. and Platen E. (1988), ‘Time discrete Taylor approximations for Itô processes with jump component’, Mathematische Nachrichten 138, 93–104.
216. Mikulevicius R. and Platen E. (1991), ‘Rate of convergence of the Euler approximation for diffusion processes’, Mathematische Nachrichten 151, 233–239.
217. Milstein G. N. (1974), ‘Approximate integration of stochastic differential equations’, Theory Probab. Appl. 19, 557–562.
218. Milstein G. N. (1978), ‘A method of second order accuracy integration of stochastic differential equations’, Theory Probab. Appl. 23, 396–401.
219. Milstein G. N. (1985), ‘Weak approximation of solutions of systems of stochastic differential equations’, Theory Probab. Appl. 30, 750–766.
220. 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.
221. Milstein G. N. (1988 a), Numerical Integration of Stochastic Differential Equations, Urals Univ. Press, Sverdlovsk. In Russian.
222. 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.
223. Milstein G. N. (1995 a), Numerical Integration of Stochastic Differential Equations, Mathematics and its Applications, Kluwer, Dordrecht/Boston/London.
224. Milstein G. N. (1995 b), ‘The solving of boundary value problems by numerical integration of stochastic equations’, Math. Comput. Simul. 38, 77–85.
225. 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.
226. 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.
227. Milstein G. N. (1997), ‘Weak approximation of a diffusion process in a bounded domain’, Stochastics and Stochastics Reports 62, 147–200.
228. 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.
229. Milstein G. N. and Tretjakov M. V. (1994), ‘Numerical solution of differential equations with colored noise’, J. Statist. Phys. 77, 691–715.
230. 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.
231. Milstein G. N., Platen E. and Schurz H. (1998), ‘Balanced implicit methods for stiff stochastic systems’, SIAM J. Numer. Anal. 35, 1010–1019.
232. Morgan B. J. (1984), Elements of Simulation, Chapman & Hall, London.
233. Mori M. (1998), ‘Low discrepancy sequences generated by piecewise linear maps’, Monte Carlo Methods Appl. 4, 141–162.
234. Müller-Gronbach T. (1996), ‘Optimal design for approximating the path of a stochastic process’, J. Statist. Planning Inf. 49, 371–385.
235. Nakazawa H. (1990), ‘Numerical procedures for sample structures on stochastic differential equations’, J. Math. Phys. 31, 1978–1990.
236. Newton N. J. (1986 a), ‘An asymptotic efficient difference formula for solving stochastic differential equations’, Stochastics 19, 175–206.
237. 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.
238. Newton N. J. (1990), ‘An efficient approximation for stochastic differential equations on the partition of symmetrical first passage times’, Stochastics 29, 227–258.
239. Newton N. J. (1991), ‘Asymptotically efficient Runge–Kutta methods for a class of Itô and Stratonovich equations’, SIAM J. Appl. Math. 51, 542–567.
240. Newton N. J. (1994), ‘Variance reduction for simulated diffusions’, SIAM J. Appl. Math. 54, 1780–1805.
241. Newton N. J. (1996), ‘Numerical methods for stochastic differential equations’, Z. Angew. Math. Mech. 76, 211–214. Suppl. 3, I–XVI.
242. 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.
243. Niederreiter H. (1988), ‘Remarks on nonlinear pseudo random numbers’, Metrika 35, 321–328.
244. Niederreiter H. (1992), Random Number Generation and Quasi-Monte-Carlo Methods, SIAM, Philadelphia, PA.
245. 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.
246. 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.
247. Ogawa S. (1992), ‘Monte Carlo simulation of nonlinear diffusion processes’, Japan J. Industrial and Appl. Math. 9, 25–33.
248. Ogawa S. (1994), ‘Monte Carlo simulation of nonlinear diffusion processes II’, Japan J. Industrial and Appl. Math. 2, 31–45.
249. Ogawa S. (1995), ‘Some problems in the simulation of nonlinear diffusion processes’, Math. Comput. Simul. 38, 217–223.
250. Ogorodnikov V. A. and Prigarin S. M. (1996), Numerical Modelling of Random Processes and Fields: Algorithms and Applications, VSP, Utrecht.
251. Oksendahl B. (1985), Stochastic Differential Equations, Springer.
252. Pardoux E. and Talay D. (1985), ‘Discretization and simulation of stochastic differential equations’, Acta Appl. Math. 3, 23–47.
253. Paskov S. and Traub J. (1995), ‘Faster valuation of financial derivatives’, J. Portfolio Manag., pp. 113–120.
254. 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.
255. Petersen W. P. (1988), ‘Some vectorized random number generators for uniform, normal and Poisson distributions for CRAY X-MP’, J. Supercomputing 1, 318–335.
256. Petersen W. P. (1994 a), ‘Lagged Fibonacci series random number generators for the NEC SX-3’, Intern. J. High Speed Computing 6, 387–398.
257. Petersen W. P. (1994 b), ‘Some experiments on numerical simulations of stochastic differential equations and a new algorithm’, J. Comput. Phys. 113, 75–81.
258. Petersen W. P. (1998), ‘A general implicit splitting for stabilizing numerical simulations of Itô stochastic differential equations’, SIAM J. Numer. Anal. 35, 1439–1451.
259. Petterson R. (1995), ‘Approximations for stochastic differential equations with reflecting convex boundaries’, Stochastic Process. Appl. 59, 295–308.
260. 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.
261. Platen E. (1980 b), ‘Weak convergence of approximations of Itô integral equations’, Z. Angew. Math. Mech. 60, 609–614.
262. Platen E. (1981), ‘An approximation method for a class of Itô processes’, Lietuvos Matem. Rink. 21, 121–133.
263. Platen E. (1982 a), ‘An approximation method for a class of Itô processes with jump component’, Lietuvos Matem. Rink. 22, 124–136.
264. Platen E. (1982 b), ‘A generalized Taylor formula for solutions of stochastic differential equations’, SANKHYA A 44, 163–172.
265. Platen E. (1983), ‘Approximation of first exit times of diffusions and approximate solution of parabolic equations’, Mathematische Nachrichten 111, 127–146.
266. Platen E. (1984), Zur zeitdiskreten Approximation von Itoprozessen, Diss. B., IMath, Akad. der Wiss. der DDR, Berlin.
267. 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.
268. 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.
269. Platen E. (1992), ‘Higher-order weak approximation of Itô diffusions by Markov chains’, Probability in the Engineering and Information Sciences 6, 391–408.
270. Platen E. (1995), ‘On weak implicit and predictor-corrector methods’, Math. Comput. Simul. 38, 69–76.
271. Platen E. and Rebolledo R. (1985), ‘Weak convergence of semimartingales and discretization methods’, Stochastic Process. Appl. 20, 41–58.
272. Platen E. and Wagner W. (1982), ‘On a Taylor formula for a class of Itô processes’, Probability and Math. Statistics 3, 37–51.
273. Protter P. (1985), ‘Approximations of solutions of stochastic differential equations driven by semimartingales’, Ann. Probab. 13, 716–743.
274. Protter P. (1990), Stochastic Integration and Differential Equations, Springer.
275. Protter P. and Talay D. (1997), ‘The Euler scheme for Levy driven stochastic differential equations’, Ann. Probab. 25, 393–423.
276. 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.
277. Rao N. J., Borwankar J. D. and Ramkrishna D. (1974), ‘Numerical solution of Itô integral equations’, SIAM J. Control Optimiz. 12, 124–139.
278. Razevig V. D. (1980), ‘Digital modelling of multi-dimensional dynamics under random perturbations’, Autom. Remote Control 4, 177–186. In Russian.
279. Ripley B. D. (1983 a), ‘Computer generation of random variables: A tutorial letter’, Inter. Statist. Rev. 45, 301–319.
280. Ripley B. D. (1983 b), Stochastic Simulation, Wiley, New York.
281. 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.
282. Ross S. M. (1991), A Course in Simulation, MacMillan, New York.
283. Rubinstein R. Y. (1981), Simulation and the Monte Carlo Method, Wiley.
284. Rümelin W. (1982), ‘Numerical treatment of stochastic differential equations’, SIAM J. Numer. Anal. 19, 604–613.
285. Ryashko L. B. and Schurz H. (1997), ‘Mean square stability analysis of some linear stochastic systems’, Dyn. Sys. Appl. 6, 165–189.
286. 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.
287. Saito Y. and Mitsui T. (1993 a), ‘Simulation of stochastic differential equations’, Ann. Inst. Statist. Math. 45, 419–432.
288. Saito Y. and Mitsui T. (1993 b), ‘T-stability of numerical schemes for stochastic differential equations’, World Sci. Ser. Appl. Anal. 2, 333–344.
289. Saito Y. and Mitsui T. (1995), ‘ S-series in the Wong-Zakai approximation for stochastic differential equations’, Vietnam J. Math. 23, 303–317.
290. Saito Y. and Mitsui T. (1996), ‘Stability analysis of numerical schemes for stochastic differential equations’, SIAM J. Numer. Anal. 33, 2254–2267.
291. 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.
292. 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.
293. Schurz H. (1996 b), ‘Numerical regularization for SDEs: Construction of nonnegative solutions’, Dyn. Sys. Appl. 5, 323–351.
294. Schurz H. (1996 c), Stability, stationarity and boundedness of some implicit numerical methods for stochastic differential equations, PhD thesis, Humboldt University, Berlin.
295. Shimizu A. and Kawachi T. (1984), ‘Approximate solutions of stochastic differential equations’, Bull. Nagoya Inst. Tech. 36, 105–108.
296. Shinozuka M. (1971), ‘Simulation of multivariate and multidimensional random differential processes’, J. Acoust. Soc. Amer. 49, 357–367.
297. 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.
298. Sloan I. H. and Wozniakowski H. (1998), ‘When are quasi-Monte-Carlo algorithms efficient for high dimensional integrals?’, J. Complexity 14, 1–33.
299. Slominski L. (1994), ‘On approximation of solutions of multidimensional SDEs with reflecting boundary conditions’, Stochastic Process. Appl. 50, 197–219.
300. 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.
301. 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.
302. Stoer J. and Bulirsch R. (1993), Introduction to Numerical Analysis, 2nd edn, Springer. (1st edn (1980).)
303. Sugita H. (1995), ‘Pseudo-random number generator by means of irrational rotation’, Monte Carlo Methods Appl. 1, 35–57.
304. Sun M. and Glowinski R. (1994), ‘Pathwise approximation and simulation for the Zakai filtering equation through operator splitting’, Calcolo 30, 219–239.
305. 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.
306. Talay D. (1982 a), Analyse Numérique des Equations Différentielles Stochastiques, PhD thesis, Université de Provence, Centre Saint Charles. Thèse 3ème Cycle.
307. 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.
308. 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.
309. Talay D. (1983 b), ‘Résolution trajectorielle et analyse numérique des équations différentielles stochastiques’, Stochastics 9, 275–306.
310. 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.
311. 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.
312. 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.
313. 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.
314. Talay D. (1991), ‘Approximation of upper Lyapunov exponents of bilinear stochastic differential equations’, SIAM J. Numer. Anal. 28, 1141–1164.
315. 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.
316. Talay D. and Tubaro L. (1990), ‘Expansion of the global error for numerical schemes solving stochastic differential equations’, Stoch. Anal. Appl. 8, 483–509.
317. 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.
318. Tezuka S. (1993), ‘Polynomial arithmetic analogue of Halton sequences’, ACM Trans. Model. Comput. Simul. 3, 99–107.
319. Tezuka S. and Tokuyama T. (1994), ‘A note on polynomial arithmetic analogue of Halton sequences’, ACM Trans. Model. Computer Simul. 4, 279–284.
320. Törok C. (1994), ‘Numerical solution of linear stochastic differential equations’, Comput. Math. Appl. 27, 1–10.
321. Traub J. F., Wasilkowski G. W. and Wozniakowski H. (1988), Information-Based Complexity, Academic Press, New York.
322. Tudor C. (1989), ‘Approximation of delay stochastic equations with constant retardation by usual Itô equations’, Rev. Roumaine Math. Pures Appl. 34, 55–64.
323. 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.
324. Tudor C. and Tudor M. (1987), ‘On approximation of solutions for stochastic delay equations’, Stud. Cerc. Mat. 39, 265–274.
325. Tudor C. and Tudor M. (1995), ‘Approximation schemes for Itô-Volterra stochastic equations’, Bol. Soc. Mat. Mexicana (3) 1, 73–85.
326. Tudor C. and Tudor M. (1997), ‘Approximate solutions for multiple stochastic equations with respect to semimartingales’, Z. Anal. Anwendungen 16, 761–768.
327. Tudor M. (1992), ‘Approximation schemes for two-parameter stochastic equations’, Probability and Math. Statistics 13, 177–189.
328. Tuffin B. (1996), ‘On the use of low discrepancy sequences in Monte Carlo methods’, Monte Carlo Methods Appl. 2, 295–320.
329. Tuffin B. (1997), ‘Comments on “On the use of low discrepancy sequences in Monte Carlo methods”’, Monte Carlo Methods Appl. 4, 87–90.
330. Unny T. E. (1984), ‘Numerical integration of stochastic differential equations in catchment modelling’, Water Res. 20, 360–368.
331. Valkeila E. (1991), ‘Computer algebra and stochastic analysis’, CWI 4, 229–238.
332. 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.
333. Wagner W. (1987), ‘Unbiased Monte-Carlo evaluation of certain functional integrals’, J. Comput. Phys. 71, 21–33.
334. 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.
335. Wagner W. (1988 b), ‘Unbiased multi-step estimators for the Monte-Carlo evaluation of certain functionals’, J. Comput. Phys. 79, 336–352.
336. Wagner W. (1989 a), Stochastische numerische Verfahren zur Berechnung von Funktionalintegralen, Habilitation, Report 02/89, IMATH, Berlin.
337. Wagner W. (1989 b), ‘Unbiased Monte-Carlo estimators for functionals of weak solutions of stochastic differential equations’, Stochastics and Stochastics Reports 28, 1–20.
338. Wagner W. and Platen E. (1978), Approximation of Itô integral equations, Preprint ZIMM, Akad. Wissenschaften, DDR, Berlin.
339. Werner M. J. and Drummond P. D. (1997), ‘Robust algorithms for solving stochastic partial differential equations’, J. Comput. Phys. 132, 312–326.
340. Wiener N. (1923), ‘Differential space’, J. Math. Phys. 2, 131–174.
341. Wong E. and Zakai M. (1965), ‘On the convergence of ordinary integrals to stochastic integrals’, Ann. Math. Statist. 36, 1560–1564.
342. Wozniakowski H. (1991), ‘Average case complexity of multivariate integration’, Bull. Amer. Math. Soc. 24, 185–194.
343. Wright D. J. (1974), ‘The digital simulation of stochastic differential equations’, IEEE Trans. Automat. Control 19, 75–76.
344. Wright D. J. (1980), ‘Digital simulation of Poisson stochastic differential equations’, Intern. J. Systems. Sci, 11, 781–785.
345. Xu K. (1995), ‘Stochastic pitchforkbifurcation: numerical simulations and symbolic calculations using Maple’, Math. Comput. Simul. 38, 199.
346. Yakowitz S. J. (1977), Computational Probability and Simulation, Addison Wesley, Reading, MA.
347. Yamada T. (1976), ‘Sur l'approximation des solutions d'équations différentielles stochastiques’, Z. Wahrsch. Verw. Gebiete 36, 153–164.
348. 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.
349. Yen Y. Y. (1988), ‘A stochastic Taylor formula for functionals of two-parameter semimartingales’, Acta Vietnamica 13, 45–54.