Probability with Martingales

Introduction – please read!

The purpose of this chapter is to give some indication of some of the ways in which the theory which we have developed can be applied to real-world problems. We consider only very simple examples, but at a lively pace!

In Sections 15.1-15.2, we discuss a trivial case of a celebrated result from mathematical economics, the Black-Scholes option-pricing formula. The formula was developed for a continuous-parameter (diffusion) model for stock prices; see, for example, Karatzas and Schreve (1988). We present an obvious discretization which also has many treatments in the literature. What needs to be emphasized is that in the discrete case, the result has nothing to do with probability, which is why the answer is completely independent of the underlying probability measure. The use of the ‘martingale measure’ ℙ in Section 15.2 is nothing other than a device for expressing some simple algebra/combinatorics. But in the diffusion setting, where the algebra and combinatorics are no longer meaningful, the martingale-representation theorem and Cameron-Martin-Girsanov change-of-measure theorem provide the essential language. I think that this justifies my giving a ‘martingale’ treatment of something which needs only junior-school algebra.

Sections 15.3-15.5 indicate the further development of the martingale formulation of optimality in stochastic control, at which Exercise E10.2 gave a first look. We consider just one ‘fun’ example, the ‘Mabinogion sheep problem’; but it is an example which illustrates rather well several techniques which may be effectively utilized in other contexts.