Stochastic Integration

Abstract

This chapter is a self-contained introduction to the theory of stochastic integration with respect to continuous semimartingales. Apart from some simple facts on the predictable σ-field (which are included with proofs in Section 1.2), we do not need any other result from the general theory of processes. The Doob—Meyer decomposition of the square of a continuous (local) martingale is given by constructing the quadratic variation process directly. Once this is done, the stochastic integral is defined and its properties are obtained in a natural fashion, using arguments common in the theory of the Lebesgue integral, namely, the monotone class theorem and dominated convergence theorem. We first give the definition of the stochastic integral with respect to Brownian motion and then deal with the case of continuous semimartingales in stages. In the last section, we define the integral with respect to a general (r.c.l.l.) semimartingale. However, this time we state without proof some key results, including the Doob—Meyer decomposition. We show that these results, three to be precise, enable us to define the integral in the general case and derive the properties of the integral following the route adopted earlier.