A New Approximation Method for Solving Stochastic Differential Equations

Abstract

We present the stochastic quadratic polynomial based method, a novel solution method for Itô stochastic differential equations (SDEs). The idea is based on numerically computing the unknown function at the next two time points, and iteratively continuing this process until the final time is reached. To achieve this, the time interval is subdivided into smaller sub-intervals, and quadratic polynomials are used to approximate the solution between two successive intervals. The main properties of the stochastic numerical methods, e.g. convergence, consistency, and stability are analyzed. We test the proposed method in an SDE problem, demonstrating promising results. We also compare our method with classic stochastic schemes, such as Euler-Maruyama (EM) and Milstein schemes, and demonstrate that the proposed method achieves higher accuracy.