An introduction to stochastic integration with respect to continuous semimartingales

Contents Preface v 1 Continuous-time stochastic processes 1 1.1 Measurability and stopping times. . Bibliography 133 iv CONTENTS Preface This monograph concerns itself with the theory of continuous-time martingales with continuous paths and the theory of stochastic integration with respect to continuous semimartingales. To set the scene for the theory to be developed, we consider an example. Assume given a probability space (Ω, F, P) endowed with a Brownian motion W , and consider a continuous mapping f : [0, ∞) → R. We would like to understand whether it is possible to define an integral t 0 f (s) dW s in a fashion analogous to ordinary Lebesgue integrals. In general, there is a correspondence between bounded signed measures on [0, t] and mappings of finite variation on [0, t]. Therefore, if we seek to define the integral with respect to Brownian motion in a pathwise sense, that is, by defining t 0 f (s) dW (ω) s for each ω separately, by reference to ordinary Lebesgue integration theory, it is necessary that the sample paths W (ω) have finite variation. However, Brownian motion has the property that its sample paths are almost surely of infinite variation on all compact intervals. Our conclusion is that the integral of f with respect to W cannot in general be defined pathwisely by immediate reference to Lebesgue integration theory. We are thus left to seek an alternate manner of defining the integral. A natural starting point is to consider Riemann sums of the form 2 n k=1 f (t n k−1)(W t n k − W t n k−1), where t n k = kt2 −n , and attempt to prove their convergence in some sense, say, in L 2. By the completeness of L 2 , it suffices to prove that the sequence of Riemann sums constitute a Cauchy sequence in L 2. In order to show this, put η(β) = sup{|f (s) − f (u)| | s, u ∈ [0, t], |s − u| ≤ β}. η(β) is called the modulus of continuity for f over [0, t]. As f is continuous, f is uniformly continuous on [0, t], and therefore η(β) tends to zero as β tends to zero. Now note that for m ≥ n, with ξ mn k