zbMATH — the first resource for mathematics

Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems. (English) Zbl 1128.65007
The authors generalize the current theory of optimal strong convergence rates for implicit Euler-based methods by allowing for Poisson-driven jumps in a stochastic differential equation. The analysis in the paper exploits a relation between backward and explicit Euler methods.

65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
Full Text: DOI
[1] Cont, R.; Tankov, P., Financial modelling with jump processes, (2004), Chapman and Hall/CRC FL · Zbl 1052.91043
[2] Gardoń, A., The order of approximation for solutions of Itô-type stochastic differential equations with jumps, Stochast. anal. appl., 22, 679-699, (2004) · Zbl 1056.60065
[3] Gikhman, I.I.; Skorokhod, A.V., Stochastic differential equations, (1972), Springer Berlin · Zbl 0169.48702
[4] Glasserman, P., Monte Carlo methods in financial engineering, (2003), Springer Berlin
[5] Higham, D.J.; Kloeden, P.E., Numerical methods for nonlinear stochastic differential equations with jumps, Numer. math., 101, 101-119, (2005) · Zbl 1186.65010
[6] Higham, D.J.; Kloeden, P.E., Convergence and stability of implicit methods for jump-diffusion systems, Internat. J. numer. anal. modeling, 3, 125-140, (2006) · Zbl 1109.65007
[7] Higham, D.J.; Mao, X.; Stuart, A.M., Strong convergence of Euler-like methods for nonlinear stochastic differential equations, SIAM J. numer. anal., 40, 1041-1063, (2002) · Zbl 1026.65003
[8] Kloeden, P.E.; Platen, E., Numerical solution of stochastic differential equations, (1999), Springer Berlin · Zbl 0701.60054
[9] Maghsoodi, Y., Mean square efficient numerical solution of jump-diffusion stochastic differential equations, Indian J. statist., 58, 25-47, (1996) · Zbl 0881.60057
[10] Maghsoodi, Y., Exact solutions and doubly efficient approximations and simulation of jump-diffusion ito equations, Stochast. anal. appl., 16, 1049-1072, (1998) · Zbl 0920.60041
[11] Maghsoodi, Y.; Harris, C.J., In-probability approximation and simulation of non-linear jump-diffusion stochastic differential equations, IMA J. math. control inform., 4, 65-92, (1987) · Zbl 0621.60064
[12] Sobczyk, K., Stochastic differential equations with applications to physics and engineering, (1991), Kluwer Academic Dordrecht · Zbl 0762.60050
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.