Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 37

CHANGE OF MEASURE

The notion of change of measure has important applications in mathematical finance. In this chapter, we present some relevant concepts and theorems such as the Girsanov theorems.

37.1 Basic Concepts and Facts

Definition 37.1 (Exponential Process). Let . The exponential process given by h is defined to be

where the B_t is a standard Brownian motion.

Theorem 37.1 (Novikov’s Theorem). Let {X_t : 0 ≤ t ≤ T} be a martingale, where T ≤ ∞. Let {M_t : 0 ≤ t ≤ T}be defined as

where X_t is the compensator of X²_t (see Theorem 34.2). Suppose that

Then {M_t : 0 ≤ t ≤ T} is a continuous martingale.

Theorem 37.2 (Lévy Characterization of Brownian Motion). Let X = {X_t = (X_t⁽¹⁾,…, X^(d)_t) : t ≥ 0} be a continuous stochastic process on the probability space with the state space (R