Introductory Stochastic Analysis for Finance and Insurance

CHAPTER 5

STOCHASTIC CALCULUS: BASIC TOPICS

5.1   STOCHASTIC (ITO) INTEGRATION

The building block of stochastic calculus is stochastic integration with respect to standard Brownian motion¹. Unlike deterministic calculus which deals with differentiation and integration of deterministic functions, stochastic calculus focuses on integration of stochastic processes. This is due in part to the nondifferentiability of Brownian motion. As we have seen in Chapter 4, the path of Brownian motion takes sharp turns everywhere and thus is nowhere differentiable. As a result, many stochastic processes which are driven by Brownian motion are also nowhere differentiable. The extension of deterministic integration to stochastic integration is not trivial, particularly when the integrand is also a stochastic process. This is because the path of Brownian motion does not have bounded variation², a basic requirement for the existence of a Riemann-Stieltjes integral. The complex nature of Brownian motion forces us to approach stochastic integration differently from deterministic integration. Instead of evaluating a stochastic integral on a path-by-path basis³, we view the Riemann-Stieltjes sums⁴ corresponding to the stochastic integral as a sequence of random variables and examine under what condition this sequence will converge (in some sense to be clarified later) to the stochastic integral. However, there are some technical issues that need to be resolved.

In what follows, we lay out the steps ...