Chapter 14 highlights that the solution to a stochastic differential equation cannot be found by treating the sample paths of stochastic processes as smooth functions of time. This is because of the nonzero quadratic variation of a Brownian motion. The purpose of Ito’s lemma is to account for this phenomenon and provides the expression of the differential of a function of a Brownian motion. This is explained using a Taylor expansion and is generalized along various dimensions. The martingale representation theorem highlights that, in some circumstances, any non-negative martingale takes the form of an exponential martingale. Therefore, in such a framework, Girsanov’s theorem does not only work for `special’ measure changes (where the Radon–Nikodym derivative process (RNDP) would coincide with an exponential martingale), but actually encompasses every equivalent measure. We establish a formal link between driftless differential equations and the martingale property of its solution. The RNDP process leading to the risk-neutral measure seen in Chapter 11 is the exponential martingale whose coefficient is such that the drift of the stock price becomes equal to the risk-free rate.

The Girsanov theorem is proved and the martingale representation theorem is presented. We again study the relation between SDEs and PIDEs, but now in the framework of a marked point process.