Milstein scheme for the numerical solution of first-order uncertain stochastic differential equations in stock price simulation

Document Type : Research Articles

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Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

10.22080/cjms.2025.29406.1763

Abstract

Uncertain stochastic calculus is a developing branch of mathematics that aims to create models incorporating both aleatory (random) and epistemic (knowledge-based) uncertainties within dynamic systems. In essence, it recognizes two types of uncertainty related to dynamical systems: randomness and belief degree. The uncertain stochastic differential equation (USDE) models such systems by simultaneously integrating randomness and human uncertainty, expressed as belief degree. This growing field has introduced a novel class of equations called USDEs. Since finding exact or analytical solutions to these equations is often difficult, numerical methods provide a practical alternative for approximating solutions. This paper investigates the use of the Milstein method for solving USDEs. Specifically, the Milstein scheme is employed to address a stock pricing problem, and its results are compared with those obtained through the fourth-order Runge-Kutta and Euler methods. The results indicate that the Milstein method yields subtle estimates of stock prices.

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