# 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 X*^{2}_{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**

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