Stochastic Calculus Honours Project Report

Preface The importance of the martingale concept cannot be overemphasized. .. . ] Martingales, Markov dependence and stationarity are the only three dependence concepts so far isolated which are suuciently general and suuciently amenable to investigation yet with a great number of deep properties. This project report deals with the stochastic calculus of semimartingales. In Doob's classical book Doo53], the semimartingales are processes which we nowadays call submartingales. The change in terminology was heralded in the preface of Lo eve's third edition (1962) of his book Probability Theory Lo e77]. A quarter of a century after Doob's book, the term semimartingale reappears on the scene, this time in Meyer's classic \Course on Stochastic Inte-grals" Mey76]. Had Lo eve written the lines at the top of this page a few years later, he would have doubtlessly included semimartingales in his list. Historically speaking, there are two ways of dealing with stochastic processes. One is via their transition probabilities, and the other is by directly dealing with their sample paths. The rst approach was favoured in the early days; researchers in stochastic processes routinely deal with paths which would horrify a typical nineteenth-century analyst! With the advent of Stochastic Calculus however, the sample path approach gained currency. The rst part of this report deals with the construction of the stochastic integral. We look at what it means to write Z b a H s dX s i when both H and X are random functions of s : stochastic processes. Wiener was the rst on the scene; having spectacularly constructed a mathematical model of Brownian motion in Wie23], he went on to deene the integral when H is an ordinary deterministic function and X is Brownian motion. The interpretation is as one would expect: a weighted sum (by H) of small increments of X. To do this, he surmounted a fundamental diiculty: the paths of X are of unbounded variation on every time interval, so that the integral above is most deenitely not a Riemann-Stieltjes integral. But the real breakthrough came with It^ o (see It^ o51]) who recognized that to integrate random functions H, the process H must not be allowed to anticipate the process X. It^ o went on to prove his famous theorem, a kind of fundamental theorem of calculus for his integral. Suddenly, stochastic integration became useful. By interpreting the Brow-nian motion as the cumulative eeect of many …