Stochastic Calculus

Abstract

We present here very briefly the basic facts of the theory of stochastic integration in the case where the integrator is a Brownian motion. Let (Ω, F, {F t }, P) be a probability space satisfying the usual conditions (see No. I.3) and W an.F t -Brownian motion in this space. The goal is to give some meaning to stochastic integrals of the type $$ {\int_0^t {{W_s}dW} _s} $$ (a) Because t ↦ W t is of infinite variation on any interval, it is not possible to define such integrals by using classical approaches from the theory of integration. This difficulty was overcome by K. Itô (1944). His far-reaching basic idea is that stochastic integrals of the type (a) can be defined via an isometry. The notion to which this approach leads us is called the Itô integral and the theory is called the Itô calculus. In particular, using the differentiation rule in the Itô calculus (see No. 7), it is seen that for all t ≥ 0 $$ \int_0^t {2{W_s}d{W_s} = W_t^2} - t\;a.s. $$ In this chapter the words “a.s.” in the statements of the above kind are usually omitted.