Three Books on Stochastic Integration

Abstract

What do the terms and calculus connote? In the past they would refer to the Ito integral, a mathematically rigorous way to make sense of with respect to a Brownian motion. Times have changed. Now that the Ito integral has become a familiar if not ubiquitous object, stochastic integration has come to refer to an imposing panoply of abstract French probability theory, only now becoming accessible to the nonspecialist. The of today, of course, has its roots in Brownian motion. The Wiener process, the mathematical model of Brownian motion, is indeed the wellspring of much of modem probability theory, perhaps due to its triple role of martingale, strong Markov process, and Gaussian process. It is the interplay of the martingale and Markov process properties that underlies the history of integration. By developing his integral in 1944 with processes as integrands, It6 [11] was able to study multidimensional diffusions with purely probabilistic techniques, an improvement over the analytic methods of Feller. Many diffusions can be represented as solutions of systems of differential equations of the form: