Numerical solution of stochastic differential equations with constant diffusion coefficients

Author:
Chien Cheng Chang

Journal:
Math. Comp. **49** (1987), 523-542

MSC:
Primary 65U05

DOI:
https://doi.org/10.1090/S0025-5718-1987-0906186-6

MathSciNet review:
906186

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present Runge-Kutta methods of high accuracy for stochastic differential equations with constant diffusion coefficients. We analyze 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.

**[1]**Ludwig Arnold,*Stochastic differential equations: theory and applications*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Translated from the German. MR**0443083****[2]**S. W. Benson,*The Foundations of Chemical Kinetics*, McGraw-Hill, New York, 1980.**[3]**S. Chandrasekhar, "Stochastic problems in physics and astronomy,"*Noise and Stochastic Processes*(N. Wax, ed.), Dover, New York, 1954.**[4]**C. C. Chang,*Numerical Solution of Stochastic Differential Equations*, Ph.D. Dissertation, University of California, Berkeley, 1985.**[5]**Alexandre Joel Chorin,*Hermite expansions in Monte-Carlo computation*, J. Computational Phys.**8**(1971), 472–482. MR**0297092****[6]**Alexandre Joel Chorin,*Accurate evaluation of Wiener integrals*, Math. Comp.**27**(1973), 1–15; corrigenda, ibid. 27 (1973), 1011. MR**0329205**, https://doi.org/10.1090/S0025-5718-1973-0329205-7**[7]**Alexandre Joel Chorin,*Numerical study of slightly viscous flow*, J. Fluid Mech.**57**(1973), no. 4, 785–796. MR**0395483**, https://doi.org/10.1017/S0022112073002016**[8]**Alexandre Joel Chorin,*Lectures on turbulence theory*, Publish or Perish, Inc., Boston, Mass., 1975. Mathematics Lecture Series, No. 5. MR**0502876****[9]**Kai Lai Chung,*A course in probability theory*, 2nd ed., Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Probability and Mathematical Statistics, Vol. 21. MR**0346858****[10]**L. Fahrmeir,*Approximation von stochastischen Differentialgleichungen auf Digitalund Hybridrechnern*, Computing**16**(1976), no. 4, 359–371. MR**0405789**, https://doi.org/10.1007/BF02252084**[11]**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**, https://doi.org/10.1016/0021-9991(84)90086-X**[12]**Joel N. Franklin,*Difference methods for stochastic ordinary differential equations*, Math. Comp.**19**(1965), 552–561. MR**0193340**, https://doi.org/10.1090/S0025-5718-1965-0193340-2**[13]**C. William Gear,*Numerical initial value problems in ordinary differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR**0315898****[14]**A. H. Jazwinski,*Stochastic Processes and Filtering Theory*, Academic Press, New York, 1970.**[15]**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 and Los Angeles, 1951, pp. 171–187. MR**0044774****[16]**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****[17]**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**, https://doi.org/10.1016/0021-9991(79)90150-5**[18]**G. N. Mil′šteĭn,*Approximate integration of stochastic differential equations*, Teor. Verojatnost. i Primenen.**19**(1974), 583–588 (Russian, with English summary). MR**0356225****[19]**G. N. Mil'shtein, "A method of second-order accuracy integration of stochastic differential equations,"*Theory Probab. Appl.*, v. 23, 1978, pp. 396-401.**[20]**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**, https://doi.org/10.1002/zamm.19800601108**[21]**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****[22]**N. J. Rao, J. D. Borwanker, and D. Ramkrishna,*Numerical solution of Ito integral equations*, SIAM J. Control**12**(1974), 125–139. MR**0343367****[23]**W. Rümelin,*Numerical treatment of stochastic differential equations*, SIAM J. Numer. Anal.**19**(1982), no. 3, 604–613. MR**656474**, https://doi.org/10.1137/0719041

Retrieve articles in *Mathematics of Computation*
with MSC:
65U05

Retrieve articles in all journals with MSC: 65U05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0906186-6

Article copyright:
© Copyright 1987
American Mathematical Society