## 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**
S. W. Benson, - 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**
A. H. Jazwinski, - 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**
G. N. Mil’shtein, "A method of second-order accuracy integration of stochastic differential equations," - 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

*The Foundations of Chemical Kinetics*, McGraw-Hill, New York, 1980. S. Chandrasekhar, "Stochastic problems in physics and astronomy,"

*Noise and Stochastic Processes*(N. Wax, ed.), Dover, New York, 1954. C. C. Chang,

*Numerical Solution of Stochastic Differential Equations*, Ph.D. Dissertation, University of California, Berkeley, 1985.

*Stochastic Processes and Filtering Theory*, Academic Press, New York, 1970.

*Theory Probab. Appl.*, v. 23, 1978, pp. 396-401.

## Additional Information

- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp.
**49**(1987), 523-542 - MSC: Primary 65U05
- DOI: https://doi.org/10.1090/S0025-5718-1987-0906186-6
- MathSciNet review: 906186