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A new approximation method for solving stochastic differential equations

Faezeh Nassajian Mojarrad

TL;DR

The paper addresses efficient numerical solution of Itô stochastic differential equations by introducing a two-step method that partitions the time interval and employs quadratic interpolation to approximate the solution on successive sub-intervals. It derives explicit update rules with coefficients $\alpha_n$ and $\beta_n$ expressed in terms of $\mu$, $\sigma$, $\Delta t$, and Wiener increments, and establishes mean-square stability, consistency, and conditional convergence under a derived stability condition. Numerical experiments on a geometric Brownian motion test show favorable stability regions and error performance compared to implicit Euler–Maruyama and Milstein schemes, including scenarios with instability for certain parameter choices. The method demonstrates practical potential for stable, accurate SDE integration and offers a framework for extending quadratic-approximation strategies to more general SDEs.

Abstract

We present a novel solution method for Itô stochastic differential equations (SDEs). We subdivide the time interval into sub-intervals, then we use the quadratic polynomials for the approximation 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 SDE problem, demonstrating promising results.

A new approximation method for solving stochastic differential equations

TL;DR

The paper addresses efficient numerical solution of Itô stochastic differential equations by introducing a two-step method that partitions the time interval and employs quadratic interpolation to approximate the solution on successive sub-intervals. It derives explicit update rules with coefficients and expressed in terms of , , , and Wiener increments, and establishes mean-square stability, consistency, and conditional convergence under a derived stability condition. Numerical experiments on a geometric Brownian motion test show favorable stability regions and error performance compared to implicit Euler–Maruyama and Milstein schemes, including scenarios with instability for certain parameter choices. The method demonstrates practical potential for stable, accurate SDE integration and offers a framework for extending quadratic-approximation strategies to more general SDEs.

Abstract

We present a novel solution method for Itô stochastic differential equations (SDEs). We subdivide the time interval into sub-intervals, then we use the quadratic polynomials for the approximation 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 SDE problem, demonstrating promising results.
Paper Structure (5 sections, 2 theorems, 41 equations, 4 figures, 3 tables)

This paper contains 5 sections, 2 theorems, 41 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Stability, consistency and convergence analysis of the method
  4. Numerical results
  5. Conclusion

Key Result

Theorem 1

Under certain conditions on the parameters, The proposed method eq:method is stable in mean square for approximating the solution of stochastic diffusion equation eq:sde.

Figures (4)

  • Figure 1: Plot of the stability region (shaded area) determined by \ref{['eq:stability_condition1']} for $\sigma=0.5$.
  • Figure 2: Plot of the stability region (shaded area) for $\sigma=0.5$. Implicit EM on the left and Milstein on the right.
  • Figure 3: Approximate and analytical solution of equation \ref{['eq:test']} using the proposed method.
  • Figure 4: 10 simulations using the proposed method for $\mu=1$.

Theorems & Definitions (8)

  • Definition 3.1
  • Theorem 1
  • proof
  • Definition 3.2
  • Definition 3.3
  • Theorem 2
  • proof
  • Remark 1