Stochastic Volterra integral Equations (SVEs) are equations of the form $$Xt = X0 +\intˆt_0 a(t, s, Xs)ds + \intˆt_0 b(t, s, Xs)dWs, t \in [0, T],$$ where a and b are measurable functions and the initial condition X0 is a random variable. The second integral in the right hand side is an Itˆo integral, which is to be taken with respect to the Brownian motion Ws. The solution Xt is a random variable for each t. Euler-Maruyama and Milstein methods have been introduced for the numerical solution of SVIEs in [3,4]. In the paper [1] the more general class theta-methods for SVIEs has been introduced and the numerical stability analysis with respect to the linear and convolution test equations, has been carried out, both in the mean-square and in the asymptotic sense. Our aim is to improve the stability properties of the stochastic theta-methods for SVIEs, by suitably modifying the quadrature formula for the increment term computation, in order to inherit good stability properties form analogous numerical methods for stochastic differential equations [2]. This is a joint work with R. D’Ambrosio from University of L’Aquila and B. Paternoster from Uni- versity of Salerno [1]D. Conte, R. D’Ambrosio, B.Paternoster, On the stability of theta-methods for stochastic Volterra integral equations, Discr. Cont. Dyn. Sys. - B, doi: 10.3934/dcdsb.2018087 (2018). [2]D.J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal. 38 (3), 753–769 (2001). [3]C.H. Wen, T.S. Zhang, Rectangular method on stochastic volterra equations, Int. J. Appl. Math. Stat. 14 (J09), 12–26 (2009). [4]C.H. Wen, T.S. Zhang, Improved rectangular method on stochastic Volterra equations, J. Comput. Appl. Math. 235 (8), 2492–2501 (2011).

### Improved theta-methods for stocastic Volterra integral equations

#### Abstract

Stochastic Volterra integral Equations (SVEs) are equations of the form $$Xt = X0 +\intˆt_0 a(t, s, Xs)ds + \intˆt_0 b(t, s, Xs)dWs, t \in [0, T],$$ where a and b are measurable functions and the initial condition X0 is a random variable. The second integral in the right hand side is an Itˆo integral, which is to be taken with respect to the Brownian motion Ws. The solution Xt is a random variable for each t. Euler-Maruyama and Milstein methods have been introduced for the numerical solution of SVIEs in [3,4]. In the paper [1] the more general class theta-methods for SVIEs has been introduced and the numerical stability analysis with respect to the linear and convolution test equations, has been carried out, both in the mean-square and in the asymptotic sense. Our aim is to improve the stability properties of the stochastic theta-methods for SVIEs, by suitably modifying the quadrature formula for the increment term computation, in order to inherit good stability properties form analogous numerical methods for stochastic differential equations [2]. This is a joint work with R. D’Ambrosio from University of L’Aquila and B. Paternoster from Uni- versity of Salerno [1]D. Conte, R. D’Ambrosio, B.Paternoster, On the stability of theta-methods for stochastic Volterra integral equations, Discr. Cont. Dyn. Sys. - B, doi: 10.3934/dcdsb.2018087 (2018). [2]D.J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal. 38 (3), 753–769 (2001). [3]C.H. Wen, T.S. Zhang, Rectangular method on stochastic volterra equations, Int. J. Appl. Math. Stat. 14 (J09), 12–26 (2009). [4]C.H. Wen, T.S. Zhang, Improved rectangular method on stochastic Volterra equations, J. Comput. Appl. Math. 235 (8), 2492–2501 (2011).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4715344
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