The Mathematics of the Martingale Approach

Abstract In this chapter we present the two main mathematical results which are needed for the application of the martingale approach to pricing and hedging. We first discuss and prove the martingale representation theorem which says that in a Wiener framework, every martingale can be represented as a stochastic integral. We then discuss and prove the Girsanov Theorem which gives us control over the class of absolutely continuous measure transformations. The abstract theory is then applied to stochastic differential equations, and to maximum likelihood estimation.