ONE LINEAR ANALYTIC APPROXIMATION FOR STOCHASTIC INTEGRODIFFERENTIAL EQUATIONS

doi:10.1016/S0252-9602(10)60104-X

Abstract

Abstract:

This article concerns the construction of approximate solutions for a general stochastic integrodifferential
equation which is not explicitly solvable and whose coefficients functionally depend on Lebesgue integrals and stochastic integrals with respect to martingales. The approximate equations are linear ordinary stochastic differential equations, the solutions of which are defined on sub-intervals of an arbitrary partition of the time
interval and connected at successive division points. The closeness of the initial and approximate solutions is measured in the L_p-th norm, uniformly on the time interval. The convergence with probability one is also given.

Key words: Stochastic integrodifferential equation, linear approximation, approximate solution, L_p-convergence,
convergence with probability one

CLC Number:

60H10

Svetlana Jankovic, Dejan Ilic. ONE LINEAR ANALYTIC APPROXIMATION FOR STOCHASTIC INTEGRODIFFERENTIAL EQUATIONS[J].Acta mathematica scientia,Series B, 2010, 30(4): 1073-1085.

Trendmd

References

[1] Ikeda N, Watanabe S. Stochastic Differential Equations and Diffusion Processes. Amsterdam: North-Holland Publish Company, 1981

[2] Karatzas I, Shreve S. Brownian Motion and Stochastic calculus. Second Edition. Springer, 1991

[3] Kloeden P E, Platen E. Numerical Solution of Stochastic Differential Equations. Third ed. Berlin: Springer-Verlag, 1999

[4] Oksendal B. Stochastic Differential Equations: An Introduction with Applications. Fifth ed. Berlin: Springer-Verlag, 1998

[5] Berger M A, Mizel V J. Volterra equations with Ito integrals I. J of Integral Equations, 1980, 2: 187--245

[6] Jankovic Sv, Jovanovic M. Generalized stochastic perturbation-depending differential equations. Stochastic Analysis and Applications, 2002, 20(6): 1281--1307

[7] Jovanovic M, Jankovic Sv. On perturbed nonlinear Itô type stochastic integrodifferential equations. J Math Anal and Appl, 2002, 269: 301--316

[8] Murge M G, Pachpatte B G. On second order Ito type stochastic integrodifferential equations. Analele stiintifice ale Universitatii. I. Cuza" din Iasi, Mathematica, 1984, Tomul xxx, s.I a, 25--34

[9] Murge M G, Pachpatte B G. Explosion and asymptotic behavior of nonlinear Ito type stochastic integrodifferential equations. Kodaim Math J, 1986, 9: 1--18

[10] Murge M G, Pachpatte B G. Succesive approximations for solutions of second order stochastic
integrodifferential equations of Ito type. Indian J Pure Appl Math, 1990, 21(3): 260--274

[11] Kloeden P E, Platen E, Schurz H. Numerical Solution of SDE Throught Computer Experiments. Third ed. Berlin: Springer-Verlag, 1999

[12] Tudor C, Tudor M. Approximation schemes for Ito-Volterra stochastic equations. Bol Soc Mat Mexicana, 1995, 3(1): 73--85

[13] Atalla M A. Finite-difference approximations for stochastic differential equations//Probabilistic Methods for the
Investigation of Systems with an Infinite Number of Degrees of Freedom: Collection of Scientific Works. Inst Math Acad Science USSR, Kiev, 1986: 11--16 (in Russian)

[14] Atalla M A. On one approximating method for stochastic differential equa\-tions//Asymptotic Methods for the Theory of Stochastic Processes, Collection of Scientific Works. Inst Math Acad Science USSR, Kiev, 1987: 15--22 (in Russian)

[15] Kanagawa S. The rate of convergence for approximate solutions of stochastic differential equations. Tokyo Math J, 1989, 12: 33--48

[16] Ilic D, Jankovic Sv. L_p-approximation of solutions of stochastic integrodifferential equations. Publ Elektrotehn Fak, Ser Math, Univ Beograd, 2001, 12: 52--60

[17] Jankovic Sv, Ilic D. An analytic approximation of solutions of sto\-cha\-stic differential equations. Computers
& Math with Appl, 2004, 47: 903--912

[18] Jankovic Sv, Ilic D. An analytic approximate method for solving stochastic integrodifferential equations. J Math Anal Appl, 2006, 320: 230--245