Numerical solution of stochastic differential equations with constant diffusion coefficients

Chang, Chien Cheng

doi:10.1090/S0025-5718-1987-0906186-6

Numerical solution of stochastic differential equations with constant diffusion coefficients
HTML articles powered by AMS MathViewer

by Chien Cheng Chang PDF

Math. Comp. 49 (1987), 523-542 Request permission

Abstract:

We present Runge-Kutta methods of high accuracy for stochastic differential equations with constant diffusion coefficients. We analyze ${L_2}$ convergence of these methods and present convergence proofs. For scalar equations a second-order method is derived, and for systems a method of order one-and-one-half is derived. We further consider a variance reduction technique based on Hermite expansions for evaluating expectations of functions of sample solutions. Numerical examples in two dimensions are presented.

References

Ludwig Arnold, Stochastic differential equations: theory and applications, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Translated from the German. MR 0443083

The Foundations of Chemical Kinetics

Noise and Stochastic Processes

Numerical Solution of Stochastic Differential Equations

Alexandre Joel Chorin, Hermite expansions in Monte-Carlo computation, J. Comput. Phys. 8 (1971), 472–482. MR 297092, DOI 10.1016/0021-9991(71)90025-8
Alexandre Joel Chorin, Accurate evaluation of Wiener integrals, Math. Comp. 27 (1973), 1–15; corrigenda, ibid. 27 (1973), 1011. MR 329205, DOI 10.1090/S0025-5718-1973-0329205-7
Alexandre Joel Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785–796. MR 395483, DOI 10.1017/S0022112073002016
Alexandre Joel Chorin, Lectures on turbulence theory, Mathematics Lecture Series, No. 5, Publish or Perish, Inc., Boston, Mass., 1975. MR 0502876
Kai Lai Chung, A course in probability theory, 2nd ed., Probability and Mathematical Statistics, Vol. 21, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0346858
L. Fahrmeir, Approximation von stochastischen Differentialgleichungen auf Digitalund Hybridrechnern, Computing 16 (1976), no. 4, 359–371. MR 405789, DOI 10.1007/BF02252084
Aaron L. Fogelson, A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting, J. Comput. Phys. 56 (1984), no. 1, 111–134. MR 760745, DOI 10.1016/0021-9991(84)90086-X
Joel N. Franklin, Difference methods for stochastic ordinary differential equations, Math. Comp. 19 (1965), 552–561. MR 193340, DOI 10.1090/S0025-5718-1965-0193340-2
C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0315898

Stochastic Processes and Filtering Theory

Paul Lévy, Wiener’s random function, and other Laplacian random functions, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley-Los Angeles, Calif., 1951, pp. 171–187. MR 0044774
E. J. McShane, Stochastic differential equations and models of random processes, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 263–294. MR 0402921
F. H. Maltz and D. L. Hitzl, Variance reduction in Monte Carlo computations using multidimensional Hermite polynomials, J. Comput. Phys. 32 (1979), no. 3, 345–376. MR 544556, DOI 10.1016/0021-9991(79)90150-5
G. N. Mil′šteĭn, Approximate integration of stochastic differential equations, Teor. Verojatnost. i Primenen. 19 (1974), 583–588 (Russian, with English summary). MR 0356225

Theory Probab. Appl.

E. Platen, Weak convergence of approximations of Itô integral equations, Z. Angew. Math. Mech. 60 (1980), no. 11, 609–614 (English, with German and Russian summaries). MR 614912, DOI 10.1002/zamm.19800601108
Eckhard Platen and Wolfgang Wagner, On a Taylor formula for a class of Itô processes, Probab. Math. Statist. 3 (1982), no. 1, 37–51 (1983). MR 715753
N. J. Rao, J. D. Borwanker, and D. Ramkrishna, Numerical solution of Ito integral equations, SIAM J. Control 12 (1974), 125–139. MR 0343367
W. Rümelin, Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal. 19 (1982), no. 3, 604–613. MR 656474, DOI 10.1137/0719041