zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Introduction to the numerical analysis of stochastic delay differential equations. (English) Zbl 0971.65004
This paper concerns the numerical approximation of the strong solution of the Itô stochastic delay differential equation (SDDE) $$dX(t)=f(X(t),X(t-\tau))dt+g(X(t),X(t-\tau))dW(t),\quad t\in[0,\tau],$$ where $X(t) =\psi(t)$, $t\in [-\tau,0]$ and $W(t)$ is a Wiener process. A theorem is proved establishing conditions for convergence, in the mean-square sense, of approximate solutions obtained from explicit single-step methods. Then a SDDE version of the Euler-Maruyama method is presented and found to have order of convergence 1. The paper concludes with several figures illustrating numerical results obtained when this method is applied to an example.

65C30Stochastic differential and integral equations
34K50Stochastic functional-differential equations
34F05ODE with randomness
65H10Systems of nonlinear equations (numerical methods)
65L06Multistep, Runge-Kutta, and extrapolation methods
65L20Stability and convergence of numerical methods for ODE
Full Text: DOI
[1] Arnold, L.: Stochastic differential equations: theory and applications. (1974) · Zbl 0278.60039
[2] C.T.H. Baker, E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, MCCM Numerical Analysis Technical Report, Manchester University, ISSSN 1360--1725, 1999.
[3] Beuter, A.; Bélair, J.: Feedback and delays in neurological diseases: a modelling study using dynamical systems. Bull. math. Biol. 55, No. 3, 525-541 (1993) · Zbl 0825.92072
[4] J.M.C. Clark, The discretization of stochastic differential equations: a primer, in: H. Neunzert (Ed.), Road Vehicle Systems and Related Mathematics; Proceedings of the second DMV-GAMM Workshop, Torino, 1987, Teubner, Stuttgart, and Kluwer Academic Publishers, Amsterdam, pp. 163--179.
[5] Driver, R. D.: Ordinary and delay differential equations, applied mathematical sciences, vol. 20. (1977) · Zbl 0374.34001
[6] Eurich, C. W.; Milton, J. G.: Noise-induced transitions in human postural sway. Phys. rev. E 54, No. 6, 6681-6684 (1996)
[7] Karatzas, I.; Shreve, S. E.: Brownian motion and stochastic calculus. (1991) · Zbl 0734.60060
[8] Kloeden, P. E.; Platen, E.: Numerical solution of stochastic differential equations. (1992) · Zbl 0752.60043
[9] Mackey, M. C.; Longtin, A.; Milton, J. G.; Bos, J. E.: Noise and critical behaviour of the pupil light reflex at oscillation onset. Phys. rev. A 41, No. 12, 6992-7005 (1990)
[10] Mao, X.: Stochastic differential equations and their applications. (1997) · Zbl 0892.60057
[11] G.N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers, Dordrecht, 1995 (Translated and revised from the 1988 Russian original.) · Zbl 0810.65144
[12] Mohammed, S. E. A.: Stochastic functional differential equations. (1984) · Zbl 0584.60066
[13] Williams, D.: Probability with martingales. (1991) · Zbl 0722.60001
[14] Zennaro, M.: Delay differential equations: theory and numerics. Theory and numerics of ordinary and partial differential equations (Leicester, 1994), 291-333 (1995) · Zbl 0847.34072