Euler-Maruyama method for solving first order uncertain stochastic differential equations

Abstract

Two forms of uncertainty are identified to be associated with dynamical systems, which are randomness and belief degree. The uncertain stochastic differential equation (USDE) is used to describe dynamical systems driven simultaneously by randomness and human uncertainty (belief degree). In this paper, the Euler-Maruyama method for solving USDEs is examined. The method is used to solve a stock pricing problem and the results are compared with those of Runge Kutta of order 4. The Euler-Maruyama method yields lower stock prices, while the stock prices from the Runge Kutta method proved to converge faster to those from the analytical method. At α = 0.5 where α ∈ (0, 1), the USDE reverts to the stochastic differential equation, with the uncertain component eliminated, showing that the USDE is indeed a hybrid of the uncertain differential equation and stochastic differential equation.