Theoretical and numerical analysis of the Euler-Maruyama method for generalized stochastic Volterra integro-differential equations

Abstract In this paper, we concern with the theoretical and numerical analysis of the generalized stochastic Volterra integro-differential equations (SVIDEs). The existence, uniqueness, boundedness and Holder continuity of the analytic solutions for generalized SVIDEs are investigated. The Euler–Maruyama method for generalized SVIDEs is presented. The boundedness of the numerical solution is proved, and the strong convergence order is obtained. The theoretical results are illustrated by some numerical examples.

[1]  Peng Hu,et al.  Stability of stochastic θ-methods for stochastic delay integro-differential equations , 2011, Int. J. Comput. Math..

[2]  S. Gan,et al.  MEAN SQUARE CONVERGENCE OF STOCHASTIC θ-METHODS FOR NONLINEAR NEUTRAL STOCHASTIC DIFFERENTIAL DELAY EQUATIONS , 2011 .

[3]  Poom Kumam,et al.  FIXED POINT THEOREMS FOR CYCLIC OPERATORS WITH APPLICATION IN FRACTIONAL INTEGRAL INCLUSIONS WITH DELAYS , 2015 .

[4]  Hui Liang,et al.  Strong superconvergence of the Euler-Maruyama method for linear stochastic Volterra integral equations , 2017, J. Comput. Appl. Math..

[5]  Mingzhu Liu,et al.  Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation , 2004 .

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

[7]  Desmond J. Higham,et al.  Almost Sure and Moment Exponential Stability in the Numerical Simulation of Stochastic Differential Equations , 2007, SIAM J. Numer. Anal..

[8]  Chengming Huang,et al.  The Stochastic -Method for Nonlinear Stochastic Volterra Integro-Differential Equations , 2014 .

[9]  Andreas Neuenkirch,et al.  CONVERGENCE OF NUMERICAL METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS IN MATHEMATICAL FINANCE , 2012, 1204.6620.

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

[11]  G. N. Milstein,et al.  Numerical Integration of Stochastic Differential Equations with Nonglobally Lipschitz Coefficients , 2005, SIAM J. Numer. Anal..

[12]  Qiang Wu,et al.  Convergence and stability of balanced methods for stochastic delay integro-differential equations , 2014, Appl. Math. Comput..

[13]  Hongli Wang,et al.  Convergence and stability of the split-step backward Euler method for linear stochastic delay integro-differential equations , 2010, Math. Comput. Model..

[14]  Hermann Brunner,et al.  Stability of numerical methods for volterra integro-differential equations , 2005, Computing.

[15]  Solutions for a class of nonlinear Volterra integral and integro-differential equation using cyclic (φ,ψ,θ)-contraction , 2013 .

[16]  X. Mao,et al.  Mean square stability of stochastic Volterra integro-differential equations , 2006, Syst. Control. Lett..

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

[18]  Xuerong Mao.  Almost Sure Exponential Stability in the Numerical Simulation of Stochastic Differential Equations , 2015, SIAM J. Numer. Anal..

[19]  D. Higham.  Stochastic Ordinary Differential Equations in Applied and Computational Mathematics , 2011 .

[20]  M. Shoaib,et al.  Fixed Point Results and its Applications to the Systems of Non-linear Integral and Differential Equations of Arbitrary Order , 2016 .

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

[22]  Hui Liang,et al.  Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay , 2019, Appl. Math. Comput..

[23]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[24]  Xuerong Mao,et al.  Stability of stochastic integro differiential equations , 2000 .