Mathematical Methods for Finance: Tools for Asset and Risk Management

CHAPTER 11

Stochastic Differential Equations

In the previous chapter, we explained stochastic integrals, a mathematical concept used for defining stochastic differential equations, the subject of this chapter. Stochastic differential equations solve the problem of giving meaning to a differential equation where one or more of its terms are subject to random fluctuations. 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 economics where they are used to express laws that govern the evolution of price probability distributions, intertemporal portfolio optimization, and conditions for continuous hedging such as in the Black-Scholes option pricing model.

   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 a noise term (or random variable) in a stochastic differential equation. In most applications in financial economics, it is assumed ...