Collocation methods for nonlinear stochastic Volterra integral equations
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Collocation methods for nonlinear stochastic Volterra integral equations Xiaoli Xu1 · Yu Xiao1
· Haiying Zhang1
Received: 26 April 2020 / Revised: 18 September 2020 / Accepted: 5 October 2020 / Published online: 23 November 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract Influenced by Xiao et al. (J Integral Equations Appl 30(1):197–218, 2018), collocation methods are developed to study strong convergence orders of numerical solutions for nonlinear stochastic Volterra integral equations under the Lipschitz condition in this paper. Some properties of exact solutions are discussed. These properties include the mean-square boundedness, the Hölder condition, and conditional expectations. In addition, this paper considers the solvability, the mean-square boundedness, and strong convergence orders of numerical solutions. At last, we validate our conclusions by numerical experiments. Keywords Collocation methods · Boundedness · Hölder condition · Solvability · Strong convergence orders Mathematics Subject Classification 34K05 · 65E05
1 Introduction Stochastic Volterra integral equations (SVIEs) arise in many different fields, such as mechanics, physics, biology, finance, and social sciences. There has been an increasing interest in the investigation of SVIEs and many scholars have made some progress. Itô (1979) and Wang (2008) studied the existence and uniqueness of solutions to SVIEs. To the best of our knowledge, the exact solutions of SVIEs are hardly available. Therefore, it is important to get numerical solutions using numerical methods. It is easy to see that SVIEs without random part are well-known Volterra integral equations (VIEs). There are extensive literatures focusing numerical analysis of VIEs. For example, Communicated by Hui Liang.
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Yu Xiao [email protected] Xiaoli Xu [email protected] Haiying Zhang [email protected]
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Harbin Institute of Technology, Harbin, China
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X. Xu et al.
Conte and Paternoster (2009) studied the Multistep collocation method for VIEs. Babolian and Shamloo (2008) investigated the numerical solutions of VIEs using operational matrices of piecewise constant orthogonal functions. Moreover, many experts adopt Legendre–Gauss collocation methods, triangular function methods, and Chebyshev polynomial methods to approximate exact solutions of VIEs (see, e.g., Sheng et al. 2014; Maleknejad et al. 2010, 2007; Maleknejad and Dehbozorgi 2018). As we know, SVIEs and stochastic differential equations (SDEs) are closely related. Numerical schemes to SDEs have been well developed. There are many references about numerical approximation to SDEs. In Shekarabi et al. (2007), Shekarabi et al. considered the strong convergence theory for SDEs under globally Lipschitz continuous conditions. In Higham et al. (2002), Mao and Stuart studied the Euler-type methods for SDEs under the local Lipschitz condition, which is very influential. Recently, implicit methods have been considered to study numerical solutions to SDEs in Mao and Sz
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