Differential Equations

SERGIO M. FOCARDI, PhD

Partner, The Intertek Group

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

Abstract: In financial modeling, the goal is to be able to represent the problem at hand as a mathematical function. In a mathematical function, the dependent variable depends on one or more variables that are referred to as independent variables. In standard calculus, there are two basic operations with mathematical functions: differentiation and integration. The differentiation operation leads to derivatives. When a mathematical function has only one independent variable, then the derivative is referred to as an ordinary derivative. Typically in financial applications, the independent variable is time. The derivative of a mathematical function that has more than one independent variable (one of which is typically time) is called a partial derivative. A differential equation is an equation that contains derivatives. When it contains only an ordinary derivative, it is referred to as an ordinary differential equation; when the differential equation contains partial derivatives, the differential equation is called a partial differential equation.

In nontechnical terms, differential equations are equations that express a relationship between a function and one or more derivatives (or differentials) of that function. The highest order of derivatives included in a differential equation is referred to as its order. In financial modeling, ...