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Convergence and stability of the balanced methods for stochastic differential equations with jumps. (English) Zbl 1236.65006

It is proved that strong balanced methods are convergent in the mean square sense with order \(1/2\) when applied to jump diffusion stochastic differential equations of the form \[ dx(t)= f(x(t-))\,dt+ g(x(t-))\,dW(t)+ u(x(t-))\,dN(t),\quad t> 0,\quad x(0-)= x_0,\tag{1} \] where \(W(t)\) is an \(m\)-dimensional Brownian motion and \(N(t)\) is a scalar Poisson process. For sufficiently small stepsize, conditions are derived that guarantee mean-square stability of strong and weak balanced methods for equation (1). Numerical results for a few examples demonstrate the order \(1/2\) convergence and describe superior stability of the balanced methods over the Euler-Maruyama method.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C20 Probabilistic models, generic numerical methods in probability and statistics
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[1] DOI: 10.1007/s10543-006-0098-4 · Zbl 1116.65004
[2] DOI: 10.1016/S0377-0427(00)00475-1 · Zbl 0971.65004
[3] DOI: 10.1016/S0377-0427(00)00467-2 · Zbl 0971.65003
[4] Cao W. R., J. Harbin Inst. Technol. 3 pp 303– (2005)
[5] Chalmers G. D., Discrete Contin. Dyn. Syst. 9 pp 47– (2008)
[6] DOI: 10.1137/070699469 · Zbl 1190.65010
[7] Cont R., Financial Modelling with Jump Processes (2004) · Zbl 1052.91043
[8] Gikhman I. I., Stochastic Differential Equations (1972) · Zbl 0494.60055
[9] Glasserman P., Monte Carlo Methods in Financial Engineering (2003) · Zbl 1038.91045
[10] DOI: 10.1007/s00211-005-0611-8 · Zbl 1186.65010
[11] Higham D. J., Int. J. Numer. Anal. Model. 3 pp 125– (2006)
[12] DOI: 10.1016/j.amc.2005.02.017 · Zbl 1095.65006
[13] Maghsoodi Y., Indian J. Statist. 58 pp 25– (1996)
[14] DOI: 10.1080/07362999808809579 · Zbl 0920.60041
[15] DOI: 10.1093/imamci/4.1.65 · Zbl 0621.60064
[16] Milstein G. N., Numerical Integration of Stochastic Differential Equations (1998) · Zbl 0810.65144
[17] DOI: 10.1137/S0036142994273525 · Zbl 0914.65143
[18] Saito Y., J. World Sci. Ser. Appl. Anal. 2 pp 333– (1993)
[19] Szpruch, L. and Mao, X. R.Strong convergence of numerical methods for nonlinear stochastic differential equations under monotone conditions. Available atwww.mathstat.strath.ac.uk/downloads/publications/3lukas_szpruch.pdf · Zbl 1262.65012
[20] DOI: 10.1016/j.apnum.2008.06.001 · Zbl 1166.65003
[21] DOI: 10.1016/j.amc.2007.03.027 · Zbl 1193.65008
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