Strong convergence rate of truncated Euler-Maruyama method for stochastic differential delay equations with Poisson jumps

Abstract

We study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.

Keywords

Truncated Euler-Maruyama method / stochastic differential delay equations / Poisson jumps / rate of the convergence

References

[1] AllenEModeling with Itô Stochastic Differential Equations, 2007, Dordrecht, Springer

[2] ApplebyJ A D, GuzowskaM, KellyC, RodkinaA. Preserving positivity in solutions of discretised stochastic differential equations. Appl Math Comput, 2010, 217: 763-774

[3] ArnoldLStochastic Differential Equations: Theory and Applications, 1974, New York, John Wiley

[4] BakerC T H, BuckwarE. Numerical analysis of explicit one-step methods for stochastic delay differential equations. J Comput Math, 2000, 3: 315-335

[5] BakerC T H, BuckwarE. Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations. J Comput Appl Math, 2005, 184: 404-427

[6] BaoJ H, BöttcherB, MaoX R, YuanC G. Convergence rate of numerical solutions to SFDEs with jumps. J Comput Appl Math, 2011, 236(2): 119-131

[7] Cong Y H, Zhan W J, Guo Q. The partially truncated Euler-Maruyama method for highly nonlinear stochastic delay differential equations with Markovian switching. Int J Comput Methods, 2020, https://doi.org/10.1142/S0219876219500142

[8] DengS N, FeiW Y, LiuW, MaoX R. The truncated EM method for stochastic differential equations with Poisson jumps. J Comput Appl Math, 2019, 355: 232-257

[9] GuoQ, MaoX R, YueR X. The truncated Euler-Maruyama method for stochastic differential delay equations. Numer Algorithms, 2018, 78(2): 599-624

[10] HighamD J, KloedenP E. Numerical methods for nonlinear stochastic differential equations with jumps. Numer Math, 2005, 101: 101-119

[11] HighamD J, MaoX R, StuartA M. Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J Numer Anal, 2002, 40: 1041-1063

[12] HuL J, LiX Y, MaoX R. Convergence rate and stability of the truncated Euler-Maruyama method for stochastic differential equations. J Comput Appl Math, 2018, 337: 274-289

[13] HutzenthalerM, JentzenA, KloedenP E. Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients. Proc R Soc A, 2011, 467: 1563-1576

[14] HutzenthalerM, JentzenA, KloedenP E. Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. Ann Appl Probab, 2012, 22: 1611-1641

[15] JacobN, WangY T, YuanC G. Stochastic differential delay equations with jumps, under nonlinear growth condition. Stochastics, 2009, 81(6): 571-588

[16] KloedenP E, PlatenENumerical Solution of Stochastic Differential Equations, 1992, Berlin, Springer

[17] LanG Q, XiaF. Strong convergence rates of modified truncated EM method for stochastic differential equations. J Comput Appl Math, 2018, 334: 1-17

[18] LiuW, MaoX R. Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations. Appl Math Comput, 2013, 223: 389-400

[19] MaoW, YouS R, MaoX R. On the asymptotic stability and numerical analysis of solutions to nonlinear stochastic differential equations with jumps. J Comput Appl Math, 2016, 301: 1-15

[20] MaoX RStochastic Differential Equations and Applications, 20072nd ed.Chichester, Horwood

[21] MaoX R. The truncated Euler-Maruyama method for stochastic differential equations. J Comput Appl Math, 2015, 290: 370-384

[22] MaoX R. Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations. J Comput Appl Math, 2016, 296: 362-375

[23] MaoX R, YuanC G, ZouJ Z. Stochastic differential delay equations. J Math Anal Appl, 2005, 304(1): 296-320

[24] MilsteinG N, PlatenE, SchurzH. Balanced implicit methods for stiff stochastic system. SIAM J Numer Anal, 1998, 35: 1010-1019

[25] SabanisS. A note on tamed Euler approximations. Electron Commun Probab, 2013, 18: 1-10

[26] SabanisS. Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients. Ann Appl Probab, 2016, 26: 2083-2105

[27] SzpruchL, MaoX R, HighamD J, PanJ Z. Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model. BIT, 2011, 51: 405-425

[28] TanL, YuanC G. Convergence rates of truncated theta-EM scheme for SDDEs. Sci Sin Math, 2020, 50: 137-154 (in Chinese)

[29] WangL S, MeiC L, XueH. The semi-implicit Euler method for stochastic differential delay equation with jumps. Appl Math Comput, 2007, 192(2): 567-578

[30] ZhaoG H, LiuM Z. Numerical methods for nonlinear stochastic delay differential equations with jumps. Appl Math Comput, 2014, 233: 222-231

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Higher Education Press