10.57647/mathsci.2025.1903.15

Numerical Computation of Solutions and Truncation Error for Stochastic Differential Equations of Order 1.5

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  • Yazid Alhojilan*1,
  1. Department of Mathematics, College of Science, Qassim University, Saudi Arabia

Received: 2025-08-10

Revised: 2025-08-26

Accepted: 2025-09-03

Published in Issue 2025-09-30

Copyright (c) 2025 Yazid Alhojilan (Author)

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This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Alhojilan, Y. (2025). Numerical Computation of Solutions and Truncation Error for Stochastic Differential Equations of Order 1.5. Mathematical Sciences, 19(3). https://doi.org/10.57647/mathsci.2025.1903.15
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Abstract

A recent study proposed a groundbreaking pathwise approximation method for numerically solving stochastic differential equations (SDEs) driven by Brownian motions. This approach eliminates the necessity of simulating stochastic Itˆo integrals, denoted by Jα. Instead, these integrals are replaced with random variables that maintain the same moments under the linear term condition.
The main objective of this method is to deliver approximate solutions with an error rate of O(h3/2). This level of precision significantly exceeds the strong error rates achieved by the Euler and Milstein methods, which are O(√h) and O(h), respectively, where h represents the step size. The development of this method relies on the assumption that the diffusion process is nondegenerate and incorporates the Itˆo-Taylor expansion alongside a modified perturbation approach. In this paper, we demonstrate that the scheme achieves strong convergence in the Wasserstein distance with an order of O(h3/2) by leveraging techniques from the optimal transport theory

Keywords

  • Stochastic Differential Equations,
  • Stochastic Systems,
  • Strong Convergence,
  • Itˆo–Taylor Expansion,
  • Coupling Method,
  • High-Order Numerical Scheme,
  • Numerical Results
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