Numerical Solution of Multidimensional Stochastic Itô-Volterra Integral Equation Based on the Least Squares Method and Block Pulse Function

Abstract

In this paper, a method based on the least squares method and block pulse function is proposed to solve the multidimensional stochastic Itô-Volterra integral equation. The Itô-Volterra integral equation is transformed into a linear algebraic equation. Furthermore, the error analysis is given by the isometry property and Doob’s inequality. Numerical examples verify the effectiveness and precision of this method.

Improvement of Split-Step Forward Milstein Schemes for SODEs Arising in Mathematical Physics

“…Stochastic ordinary di erential equations (SODEs) play a pivotal role in explaining some physical phenomena such as chemical reactions [9], nancial mathematics [10], mathematical ecology [11], epidemiology [12], medicine [13], and population dynamics [14]. Generally, SODEs cannot be solved analytical, but many numerical solutions can be found, for instance, the split-step theta Milstein method [15], the least-squares method [16], the discrete Temimi-Ansari method [17], the improved Euler-Maruyama method [18], the ve-stage Milstein method [19], the split-step Milstein method [20], the split-step Adams-Moulton Milstein method [21], the split-step forward Milstein method [22], and the Runge-Kutta method [23,24].…”