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Stochastic Integrals

SERGIO M. FOCARDI, PhD

Partner, The Intertek Group

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

Abstract: Calculus is an important tool because it provides two key ideas for financial modeling: (1) the concept of instantaneous rate of change, and (2) a framework and rules for linking together quantities and their instantaneous rates of change. Calculus made the concept of infinitely small quantities precise with the notion of limit. If the rate of change can get arbitrarily close to a definite number by making the time interval sufficiently small, that number is the instantaneous rate of change. The instantaneous rate of change is the limit of the rate of change when the length of the interval gets infinitely small. This limit is referred to as the derivative of a function, or simply derivative. Starting from this definition and with the help of a number of rules for computing a derivative, it was shown that the instantaneous rate of change of a number of functions can be explicitly computed as a closed formula. The process of computing a derivative, referred to as differentiation, solves the problem of finding the steepness of the tangent to a curve; the process of integration solves the problem of finding the area below a given curve. A key result of calculus is the discovery that integration and derivation are inverse operations: Integrating the derivative of a function yields the function itself. Standard calculus deals ...

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