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Convergence rate of numerical solutions to SFDEs with jumps. (English) Zbl 1236.65005

The authors consider stochastic functional differential equations (SFDEs) with jumps of the form

dx(t)=f(x t )dt+g(x t )dB t +h(x t )dN(t),0tT,

with given x 0 , where x is n-dimensional,x t :={x(t+θ),-τθ0}, x t - :={x((t+θ) - ),-τθ0},x(t - ):=lim st x(s), B(t) is an m-dimensional Brownian motion, and N(t) is a scalar Poisson process.

Under a global Lipschitz condition they show that the pth-moment convergence of Euler-Maruyama numerical solutions to SFDEs with jumps has the order 1/p for any p2. This is different from the case of SFDEs without jumps, where the order is 1/2 for any p2. They consider also the mean-square convergence under a local Lipschitz condition.

65C30Stochastic differential and integral equations
65L20Stability and convergence of numerical methods for ODE
60H35Computational methods for stochastic equations
34K50Stochastic functional-differential equations
34K28Numerical approximation of solutions of functional-differential equations
34F05ODE with randomness
60H10Stochastic ordinary differential equations
60J65Brownian motion
[1]Hale, J. K.; Lunel, S. M. V.: Introduction to functional differential equations, (1993)
[2]Kolmanovskii, V. B.; Nosov, V. R.: Stability and periodic modes of control systems with after effect, (1981) · Zbl 0457.93002
[3]Mao, X.: Stochastic differential equations and applications, (1997)
[4]Gukhal, C. R.: The compound option approach to American options on jump-diffusions, J. econom. Dynam. control 28, 2055-2074 (2004) · Zbl 1201.91195 · doi:10.1016/j.jedc.2003.06.002
[5]Cont, R.; Tankov, P.: Financial modelling with jump processes, (2004)
[6]Sobczyk, K.: Stochastic differential equations with applications to physics and engineering, (1991) · Zbl 0762.60050
[7]Higham, D. J.; Kloeden, P. E.: Numerical methods for nonlinear stochastic differential equations with jumps, Numer. math. 101, 101-119 (2005) · Zbl 1186.65010 · doi:10.1007/s00211-005-0611-8
[8]Higham, D. J.; Kloeden, P. E.: Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. comput. Appl. math. 205, 949-956 (2007) · Zbl 1128.65007 · doi:10.1016/j.cam.2006.03.039
[9]Mao, X.: Numerical solutions of stochastic functional differential equations, LMSJ. comput. Math. 6, 141-161 (2003) · Zbl 1055.65011 · doi:http://www.lms.ac.uk/jcm/6/lms2002-027/
[10]Yuan, C.; Mao, X.: A note on the rate of convergence of the Euler–Maruyama method for scholastic differential equations, Stoch. anal. Appl. 26, 325-333 (2008) · Zbl 1136.60040 · doi:10.1080/07362990701857251
[11]Billingsley, P.: Convergence of probability measures, (1968) · Zbl 0172.21201
[12]Protter, P. E.: Stochastic integration and differential equations, (2004)
[13]Fang, S.; Imkeller, P.; Zhang, T.: Global flows for stochastic differential equations without global Lipschitz conditions, Ann. probab. 35, 180-205 (2007) · Zbl 1128.60046 · doi:10.1214/009117906000000412