Difference Equations

SERGIO M. FOCARDI, PhD

Partner, The Intertek Group

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

Abstract: The theory of linear difference equations has found applications in many areas in finance. A difference equation is an equation that involves differences between successive values of a function of a discrete variance. The theory of linear difference equations covers three areas: solving difference equations, describing the behavior of difference equations, and identifying the equilibrium (or critical value) and stability of difference equations.

Linear difference equations are important in the context of dynamic econometric models. Stochastic models in finance are expressed as linear difference equations with random disturbances added. Understanding the behavior of solutions of linear difference equations helps develop intuition about the behavior of these models. The relationship between difference equations (the subject of this entry) and differential equations is as follows. The latter are great for modeling situations in finance where there is a continually changing value. The problem is that not all changes in value occur continuously. If the change in value occurs incrementally rather than continuously, then differential equations have their limitations. Instead, a financial modeler can use difference equations, which are recursively defined sequences.

In this entry we explain the theory of linear difference equations ...

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