Title: Numerical methods for first order uncertain stochastic differential equations

Authors: Justin Chirima; Eriyoti Chikodza; Senelani Dorothy Hove-Musekwa

Addresses: Department of Mathematics and Computer Science, Great Zimbabwe University, Masvingo, Zimbabwe ' Department of Mathematics and Computer Science, Great Zimbabwe University, Masvingo, Zimbabwe ' Department of Applied Mathematics, National University of Science and Technology, Bulawayo, Zimbabwe

Abstract: Uncertain stochastic calculus is a relatively new sub discipline of mathematics. This branch of mathematical sciences seeks to develop models that capture aleatory and epistemic features of generic uncertainty in dynamical systems. The growth of uncertain stochastic theory has given birth to a novel class of differential equations called uncertain stochastic differential equations (USDEs). Exact and analytic solutions to this family of differential equations are not always available. In such cases, numerical analysis provides a gateway to approximate solutions. This paper examines a Runge-Kutta method for solving USDEs. Before examining and applying the Runge-Kutta method, the paper states and proves the existence and uniqueness theorem. The Runge-Kutta method is then applied to solve an American call option pricing problem. This numerical algorithm proves to be effective and efficient because it produces almost the same results as compared to Chen's analytic formula and the classical Black-Scholes model.

Keywords: Runge-Kutta; call option; uncertainty; aleatory; epistemic.

DOI: 10.1504/IJMOR.2020.104679

International Journal of Mathematics in Operational Research, 2020 Vol.16 No.1, pp.1 - 23

Received: 21 Jun 2018
Accepted: 22 Aug 2018
Published online: 28 Jan 2020 *

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