# An orthogonal basis expansion method for solving path-independent stochastic differential equations

## TL;DR

This paper introduces an orthogonal basis expansion method using 2D-Hermite polynomials to solve path-independent stochastic differential equations efficiently, enabling direct computation of expectation and variance.

## Contribution

The paper develops a novel orthogonal basis expansion approach for SDEs with path-independent solutions, utilizing 2D-Hermite polynomials and solving nonlinear equations for coefficients.

## Key findings

- Method accurately computes expectation and variance.
- Numerical results show high efficiency and validity.
- Outperforms some existing numerical methods.

## Abstract

In this article, we present an orthogonal basis expansion method for solving stochastic differential equations with a path-independent solution of the form $X_{t}=\phi(t,W_{t})$. For this purpose, we define a Hilbert space and construct an orthogonal basis for this inner product space with the aid of 2D-Hermite polynomials. With considering $X_{t}$ as orthogonal basis expansion, this method is implemented and the expansion coefficients are obtained by solving a system of nonlinear integro-differential equations. The strength of such a method is that expectation and variance of the solution is computed by these coefficients directly. Eventually, numerical results demonstrate its validity and efficiency in comparison with other numerical methods.