Open AccessJournal Article
Approximate solution of the stochastic Volterra integral equations via expansion method
TL;DR: In this paper, the Taylor expansion method is used to transform the second kind Volterra integral equation (SVIE) to a linear stochastic ordinary differential equation, which needs specified boundary conditions.
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Abstract: In this paper, we present an efficient method for determining the solution of the stochastic second kind Volterra integral equations (SVIE) by using the Taylor expansion method. This method transforms the SVIE to a linear stochastic ordinary differential equation which needs specified boundary conditions. For determining boundary conditions, we use the integration technique. This technique gives an approximate simple and closed form solution for the SVIE. Expectation of the approximating process is computed. Some numerical examples are used to illustrate the accuracy of the method.
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Citations
Stochastic Differential Equations
Ioannis Karatzas,Steven E. Shreve +1 more
- 01 Jan 1998
TL;DR: In this paper, the authors explore questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties, using diffusion processes as a model of a Markov process with continuous sample paths.
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Euler polynomial solutions of nonlinear stochastic Itô–Volterra integral equations
TL;DR: A practical matrix method based on operational matrices of integration and collocation points is presented to find the approximate solution of nonlinear stochastic Ito–Volterra integral equations and an upper error bound is provided under mild conditions.
54
Fractional Conformable Stochastic Integrodifferential Equations: Existence, Uniqueness, and Numerical Simulations Utilizing the Shifted Legendre Spectral Collocation Algorithm
TL;DR: In this article , a spectral collocation algorithm is proposed to transform fractional conformable stochastic integrodifferential equations into a system containing a finite number of algebraic equations that can be treated using familiar numerical methods.
An efficient approach based on radial basis functions for solving stochastic fractional differential equations
TL;DR: In this article, a collocation method based on Gaussian Radial Basis Functions (RBFs) for approximating the solution of stochastic fractional differential equations (SFDEs) is presented.
Stochastic integrodifferential models of fractional orders and Leffler nonsingular kernels: well-posedness theoretical results and Legendre Gauss spectral collocation approximations
TL;DR: In this article , a combination of sufficient conditions, topological theorems, and Banach space theory are utilized to construct the well-posedness proof for a specific form of stochastic fractional integrodifferential models considering the Leffler nonsingular kernels operator.
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References
Stochastic partial differential equations
Helge Holden,Bernt Øksendal,Jan Ubøe,Tusheng Zhang +3 more
- 01 Jan 1996
TL;DR: In this paper, the general theory developed in Chapter 2 to solve various stochastic partial differential equations (SPDEs) was applied to solve some of the basic SPDEs that appear frequently in applications.
Euler schemes and large deviations for stochastic Volterra equations with singular kernels
TL;DR: In this article, an Euler type approximation is constructed for stochastic Volterra equation with singular kernels, which provides an algorithm for numerical calculation, and the large deviation estimates of small perturbation to equations of this type are obtained.
135
Numerical solution of random differential equations: A mean square approach
TL;DR: This paper deals with the construction of numerical solutions of random initial value differential problems by means of a random Euler difference scheme whose mean square convergence is proved based on conditions expressed in terms of the mean square behavior of the right-hand side of the underlying random differential equation.
84
Numerical solution of stochastic differential equations by second order Runge-Kutta methods
TL;DR: The numerical solutions of stochastic initial value problems via random Runge-Kutta methods of the second order are proposed and mean square convergence of these methods is proved and a random mean value theorem is established.
69