This study aims to investigate a stochastic Volterra integral equation driven by fractional Brownian motion with Hurst parameter $H\in (\frac 12, 1)$. We employ the Wong-Zakai approximation to simplify this intricate problem, transforming the stochastic integral equation into an ordinary integral equation. Moreover, we consider the convergence and the rate of convergence of the Wong-Zakai approximation for this kind of equation.
[1] M. Berger and V. Mizel, Volterra equations with Itˆo integrals I, J. Integral Equ., 2(3) (1980), 187-245.
[2] M. Besal`u and C. Rovira, Stochastic Volterra equations driven by fractional Brownian motion with Hurst parameter
[3] W. Cao, ZH. Zhang, and G. E. Karniadakis, Numerical methods for delay differential equations via the WongZakai approximation, SIAM J. Sci. Comput., 37 (2015), 295-318.
[4] L. Coutin and L. Decreusefond, Abstract nonlinear filtering theory in the presence of fractional Brownian motion, Ann. Appl. Probab., 9 (1999), 1058-1090.
[5] L. Decreusefond and S. Ustu¨nel,¨ Stochastic analysis of the fractional Brownian motion, Potential Anal., 10 (1999), 177-214.
[6] N. T. Dung, Stochastic Volterra integro-differential equations driven by fractional Brownian motion in a Hilbert space, Stochs. Int. J Prob. Stoch. Process., 87 (2014), 142-159.
[7] P. Hu and Ch. Huang, The Stochastic θ-Method for Nonlinear Stochastic Volterra Integro differential equations, Abstr. Appl. Anal., (2014), 1-13.
[8] M. Kamrani and N. Jamshidi, Implicit Milstein method for stochastic differential equations via the Wong-Zakai approximation, Numer. Algorithms. 79 (2018), 357-374.
[9] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, Germany, 1992.
[10] V. Lakshmikantham, D. Bainov, and P. Simeonov, Theory of Impulsive Differential Equations, World Scientific Press, Singapore 6, 1989.
[11] M. Li, C. Huang, and Y. Hu, Numerical methods for stochastic Volterra integral equations with weakly singular kernels, IMA J. Numer. Anal., 42(3) (2022), 2656-2683.
[12] P. Protter, Volterra equations driven by semimartingales, Ann. Appl. Probab., 13 (1985), 519-530.
[13] S. Saha Ray and P. Singh, Numerical solution of stochastic Itˆo-Volterra integral equation by using Shifted Jacobi operational matrix method, Appl. Math. Comput., 410 (2021), 126440.
[14] C. Tudor, Wong-Zakai type approximations for stochastic differential equations driven by a fractional Brownian motion, J. Eur. Math. Soc., 28 (2009), 165-182.
[15] K. Twardowska, On the approximation theorem of the Wong-Zakai type for the functional stochastic differential equations, Probab. Math. Stat., 12 (1991), 319-334.
[16] K. Twardowska, T. Marnik, and M. Paslawaska-Poluniak, Approximation of the Zakai equation in a nonlinear filtering problem with delay, Int. J. Appl. Math. Comput. Sci., 13 (2003), 151-160.
[17] E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist., 36 (1965), 1560-1564.
[18] J. Wu, G. Jiang, and X. Sang, Numerical solution of nonlinear stochastic Itˆo Volterra integral equations based on Haar wavelets, Adv. Differ. Equ., 2019(1) (2019), 1-14.
[19] L. Wuan, Wiener chaos expansion and numerical solutions of stochastic partial differential equations, Dissertation (Ph.D.), California Institute of Technology, 2006.
Kamrani, M. (2024). Wong-Zakai approximation of stochastic Volterra integral equations. Computational Methods for Differential Equations, 12(3), 484-501. doi: 10.22034/cmde.2023.58696.2485
MLA
Minoo Kamrani. "Wong-Zakai approximation of stochastic Volterra integral equations". Computational Methods for Differential Equations, 12, 3, 2024, 484-501. doi: 10.22034/cmde.2023.58696.2485
HARVARD
Kamrani, M. (2024). 'Wong-Zakai approximation of stochastic Volterra integral equations', Computational Methods for Differential Equations, 12(3), pp. 484-501. doi: 10.22034/cmde.2023.58696.2485
VANCOUVER
Kamrani, M. Wong-Zakai approximation of stochastic Volterra integral equations. Computational Methods for Differential Equations, 2024; 12(3): 484-501. doi: 10.22034/cmde.2023.58696.2485