Stochastic Differential Equations

Abstract

In nontechnical terms, differential equations are equations that express a relationship between a function and one or more derivatives (or differentials) of that function. It would be difficult to overemphasize the importance of differential equations in financial modeling where they are used to express laws that govern the evolution of price probability distributions, the solution of economic variational problems (such as intertemporal optimization), and conditions for continuous hedging (such as in the Black-Scholes equation). The two broad types of differential equations are ordinary differential equations and partial differential equations. The former are equations or systems of equations involving only one independent variable; the latter are differential equations or systems of equations involving partial derivatives. When one or more of the variables is a stochastic process, we have the case of stochastic differential equations and the solution is also a stochastic process. An assumption must be made about driving noise in a stochastic differential equation. In most applications, it is assumed that the noise term follows a Gaussian random variable, although types of random variables can be assumed. Keywords: stochastic differential equation; white noise; Random walk; Brownian motion; Ito process; cannot; arithmetic Brownian motion; Ornstein-Uhlenbeck process; geometric Brownian motion