Convergence and stability of the backward Euler method for jump-diffusion SDEs with super-linearly growing diffusion and jump coefficients

Abstract This paper firstly investigates convergence of the backward Euler method for stochastic differential equations (SDEs) driven by Brownian motion and compound Poisson process. Different from existing works, we do not impose a globally Lipschitz condition on the jump coefficient but formulate a more relaxed condition to allow for its super-linear growth. It is shown that the mean square convergence order of this method can be arbitrarily close to 1 2 under mild assumptions imposed on SDEs, allowing for possibly super-linearly growing drift, diffusion and jump coefficients. An exact order 1 2 is recovered when further differentiability assumption is put on the coefficients. Furthermore, the considered method is able to inherit the mean square stability of a wider class of Levy noise driven SDEs for all stepsizes. These results are finally supported by some numerical experiments.

[1]  R. Situ.  Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering , 2005 .

[2]  X. Q. Liu,et al.  Weak Approximations and Extrapolations of Stochastic Differential Equations with Jumps , 2000, SIAM J. Numer. Anal..

[3]  Xuerong Mao,et al.  Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations , 2016, J. Comput. Appl. Math..

[4]  Jordi Villà-Freixa,et al.  Simulation methods with extended stability for stiff biochemical Kinetics , 2009, BMC Systems Biology.

[5]  L. Szpruch,et al.  Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model , 2011 .

[6]  Chengming Huang,et al.  Exponential mean square stability of numerical methods for systems of stochastic differential equations , 2012, J. Comput. Appl. Math..

[7]  A. R. Humphries,et al.  Dynamical Systems And Numerical Analysis , 1996 .

[8]  Xiaojie Wang,et al.  Exponential integrability properties of numerical approximation processes for nonlinear stochastic differential equations , 2013, Math. Comput..

[9]  Andrew M. Stuart,et al.  Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations , 2002, SIAM J. Numer. Anal..

[10]  Siqing Gan,et al.  Compensated stochastic theta methods for stochastic differential equations with jumps , 2010 .

[11]  Konstantinos Dareiotis,et al.  On Tamed Euler Approximations of SDEs Driven by Lévy Noise with Applications to Delay Equations , 2014, SIAM J. Numer. Anal..

[12]  Hui Yu,et al.  Numerical Solutions of Stochastic Differential Equations Driven by Poisson Random Measure with Non-Lipschitz Coefficients , 2012, J. Appl. Math..

[13]  E. Platen,et al.  Balanced Implicit Methods for Stiff Stochastic Systems , 1998 .

[14]  István Gyöngy,et al.  On stochastic equations with respect to semimartingales I. , 1980 .

[15]  Xuerong Mao,et al.  Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients , 2012, J. Comput. Appl. Math..

[16]  Xuerong Mao,et al.  Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients , 2013 .

[17]  Zhongqiang Zhang,et al.  A Fundamental Mean-Square Convergence Theorem for SDEs with Locally Lipschitz Coefficients and Its Applications , 2012, SIAM J. Numer. Anal..

[18]  R. Cont,et al.  Financial Modelling with Jump Processes , 2003 .

[19]  J. B. Walsh,et al.  An introduction to stochastic partial differential equations , 1986 .

[20]  Jean Jacod,et al.  The approximate Euler method for Lévy driven stochastic differential equations , 2005 .

[21]  Fuke Wu,et al.  Choice of θ and mean-square exponential stability in the stochastic theta method of stochastic differential equations , 2014, J. Comput. Appl. Math..

[22]  Raphael Kruse,et al.  Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition , 2015, 1509.00609.

[23]  K. Burrage,et al.  Numerical methods for strong solutions of stochastic differential equations: an overview , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[24]  Xu Yang,et al.  A transformed jump-adapted backward Euler method for jump-extended CIR and CEV models , 2016, Numerical Algorithms.

[25]  Nicola Bruti-Liberati.  Numerical Solution of Stochastic Differential Equations with Jumps in Finance , 2010 .

[26]  S. Sabanis.  Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients , 2013, 1308.1796.

[27]  Desmond J. Higham,et al.  Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems , 2007 .

[28]  Chaman Kumar,et al.  On Explicit Approximations for Lévy Driven SDEs with Super-linear Diffusion Coefficients , 2016, 1611.03417.

[29]  Xiaojie Wang,et al.  A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise , 2012 .

[30]  Xuerong Mao,et al.  The truncated Euler-Maruyama method for stochastic differential equations , 2015, J. Comput. Appl. Math..

[31]  C. Kumar.  Explicit numerical schemes of SDEs driven by Lévy noise with super-linear coeffcients and their application to delay equations , 2015 .

[32]  A. Gardon.  The Order of Approximations for Solutions of Itô-Type Stochastic Differential Equations with Jumps , 2004 .

[33]  Desmond J. Higham,et al.  Numerical methods for nonlinear stochastic differential equations with jumps , 2005, Numerische Mathematik.

[34]  Desmond J. Higham,et al.  Exponential mean square stability of numerical solutions to stochastic differential equations , 2003 .

[35]  Xiaojie Wang,et al.  The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients , 2011, 1102.0662.

[36]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[37]  Nicola Bruti-Liberati,et al.  Strong approximations of stochastic differential equations with jumps , 2007 .

[38]  P. Kloeden,et al.  Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients , 2010, 1010.3756.

[39]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[40]  Wolf-Jürgen Beyn,et al.  Stochastic C-Stability and B-Consistency of Explicit and Implicit Euler-Type Schemes , 2014, J. Sci. Comput..

[41]  Heping Ma,et al.  Order-preserving strong schemes for SDEs with locally Lipschitz coefficients , 2014, 1402.3708.

[42]  Lukasz Szpruch,et al.  V-integrability, asymptotic stability and comparison property of explicit numerical schemes for non-linear SDEs , 2017, Math. Comput..

[43]  M. Hutzenthaler,et al.  Numerical Approximations of Stochastic Differential Equations With Non-globally Lipschitz Continuous Coefficients , 2012, 1203.5809.

[44]  Chaman Kumar,et al.  On tamed milstein schemes of SDEs driven by Lévy noise , 2014, 1407.5347.

[45]  A. Kohatsu-Higa,et al.  Jump-adapted discretization schemes for Lévy-driven SDEs , 2010 .

[46]  P. Kloeden,et al.  CONVERGENCE AND STABILITY OF IMPLICIT METHODS FOR JUMP-DIFFUSION SYSTEMS , 2005 .

[47]  P. Kloeden,et al.  Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.