Well-posedness and EM approximations for non-Lipschitz stochastic fractional integro-differential equations

Abstract This paper considers the nonlinear stochastic fractional integro-differential equations (SFIDEs) under the non-Lipschitz conditions, which are general and include many stochastic (fractional) integro-differential equations discussed in literature. An important connection between SFIDEs and stochastic Volterra integral equations (SVIEs) is derived in detail by the Fubini theorem. Using the Euler–Maruyama (EM) approximation, we prove the existence, uniqueness and stability results of the solution to SFIDEs. Moreover, it is shown that the modified EM solution of SFIDEs shares strong first-order sharp convergence. The numerical examples are performed to show the accuracy and effectiveness of the numerical scheme and verify the correctness of our theoretical analysis.

[1]  V. E. Tarasov.  Fractional integro-differential equations for electromagnetic waves in dielectric media , 2009, 1107.5892.

[2]  Esmail Babolian,et al.  Numerical solution of stochastic fractional integro-differential equation by the spectral collocation method , 2017, J. Comput. Appl. Math..

[3]  M. R. Hooshmandasl,et al.  Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations , 2016 .

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

[5]  Farshid Mirzaee,et al.  Application of orthonormal Bernstein polynomials to construct a efficient scheme for solving fractional stochastic integro-differential equation , 2017 .

[6]  KEVIN G. TEBEEST.  Classroom Note: Numerical and Analytical Solutions of Volterra's Population Model , 1997, SIAM Rev..

[7]  Mohammad Heydari,et al.  An iterative technique for the numerical solution of nonlinear stochastic Itô -Volterra integral equations , 2018, J. Comput. Appl. Math..

[8]  J. Trujillo,et al.  On the existence of solutions of fractional integro-differential equations , 2011 .

[9]  M. Asgari,et al.  Block pulse approximation of fractional stochastic integro-differential equation , 2014 .

[10]  Chris P. Tsokos,et al.  On the Existence, Uniqueness, and Stability Behavior of a Random Solution to a Nonlinear Perturbed Stochastic Integro-Differential Equation , 1975, Inf. Control..

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

[12]  G. S. Ladde,et al.  Stochastic fractional differential equations: Modeling, method and analysis , 2012 .

[13]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[14]  Farshid Mirzaee,et al.  Euler polynomial solutions of nonlinear stochastic Itô-Volterra integral equations , 2018, J. Comput. Appl. Math..

[15]  Minoo Kamrani.  Convergence of Galerkin method for the solution of stochastic fractional integro differential equations , 2016 .

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

[17]  Ji Li,et al.  Carathéodory approximations and stability of solutions to non-Lipschitz stochastic fractional differential equations of Itô-Doob type , 2018, Appl. Math. Comput..

[18]  Khosrow Maleknejad,et al.  Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions , 2012, Math. Comput. Model..

[19]  Farshid Mirzaee,et al.  On the numerical solution of fractional stochastic integro-differential equations via meshless discrete collocation method based on radial basis functions , 2019, Engineering Analysis with Boundary Elements.

[20]  Carlo Cattani,et al.  An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics , 2015, J. Comput. Phys..

[21]  Ichiro Ito.  On the existence and uniqueness of solutions of stochastic integral equations of the Volterra type , 1979 .

[22]  Khosrow Maleknejad,et al.  Numerical solution of a stochastic population growth model in a closed system , 2013 .

[23]  F. M. Scudo,et al.  Vito Volterra and theoretical ecology. , 1971, Theoretical population biology.

[24]  Hang Xu,et al.  Analytical approximations for a population growth model with fractional order , 2009 .

[25]  E. Renshaw,et al.  STOCHASTIC DIFFERENTIAL EQUATIONS , 1974 .

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

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

[28]  V. Lakshmikantham,et al.  Theory of Integro-Differential Equations , 1995 .

[29]  Dung Nguyen Tien.  Fractional stochastic differential equations with applications to finance , 2013 .

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

[31]  F. Mohammadi.  Efficient Galerkin solution of stochastic fractional differential equations using second kind Chebyshev wavelets , 2017 .

[32]  M. Maleki,et al.  Numerical approximations for Volterra’s population growth model with fractional order via a multi-domain pseudospectral method , 2015 .

[33]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .