×

Numerical analysis of explicit one-step methods for stochastic delay differential equations. (English) Zbl 0974.65008

Summary: We consider the problem of strong approximations of the solution of stochastic differential equations of Itô form with a constant lag in the argument. We indicate the nature of the equations of interest, and give a convergence proof in full detail for explicit one-step methods. We provide some illustrative numerical examples, using the Euler-Maruyama scheme.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
34K50 Stochastic functional-differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Milstein, Numerical integration of stochastic differential equations (1995) · Zbl 0810.65144 · doi:10.1007/978-94-015-8455-5
[2] DOI: 10.1007/BF01204909 · Zbl 0574.47005 · doi:10.1007/BF01204909
[3] Driver, Ordinary and delay differential equations (1977) · Zbl 0374.34001 · doi:10.1007/978-1-4684-9467-9
[4] Crauel, Stochastic dynamics 2 (1997)
[5] Clark, Road Vehicle Systems and Related Mathematics pp 163– (1987)
[6] Burrage, Ann. Numer. Math. 1 pp 63– (1994)
[7] Beuter, Bull. Math. Biol. 55 pp 525– (1993)
[8] DOI: 10.1007/s002850050133 · Zbl 0908.92026 · doi:10.1007/s002850050133
[9] Baker, J. Comput. Appl. Math.
[10] Zennaro, Theory and numerics of ordinary and partial differential equations pp 291– (1994)
[11] Mao, Stochastic differential equations and their applications (1997) · Zbl 0892.60057
[12] DOI: 10.1103/PhysRevA.41.6992 · doi:10.1103/PhysRevA.41.6992
[13] Küchler, Discussion paper 25 pp 1436– (1999)
[14] Kolmanovskiĭ, Applied theory of functional-differential equations (1992) · doi:10.1007/978-94-015-8084-7
[15] DOI: 10.1090/S0025-5718-99-01177-1 · Zbl 0948.65002 · doi:10.1090/S0025-5718-99-01177-1
[16] DOI: 10.1137/S0036139995286515 · Zbl 0888.60046 · doi:10.1137/S0036139995286515
[17] DOI: 10.1103/PhysRevE.54.6681 · doi:10.1103/PhysRevE.54.6681
[18] Williams, Probability with martingales (1991) · Zbl 0722.60001 · doi:10.1017/CBO9780511813658
[19] Tudor, Stud. Cere. Mat. 44 pp 73– (1992)
[20] DOI: 10.1007/3-540-60214-3_51 · doi:10.1007/3-540-60214-3_51
[21] DOI: 10.1137/S0036142992228409 · Zbl 0869.60052 · doi:10.1137/S0036142992228409
[22] DOI: 10.1017/S0962492900002920 · doi:10.1017/S0962492900002920
[23] Mohammed, Stochastic analysis and related topics VI pp 1– (1996)
[24] Mohammed, Stochastic functional differential equations (1984) · Zbl 0584.60066
[25] DOI: 10.1016/S0377-0427(98)00139-3 · Zbl 0928.65015 · doi:10.1016/S0377-0427(98)00139-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.