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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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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

equation

where the Bt is a standard Brownian motion.

Theorem 37.1 (Novikov’s Theorem). Let {Xt : 0 ≤ tT} be a martingale, where T ≤ ∞. Let {Mt : 0 ≤ tT}be defined as

equation

where Xt is the compensator of X2t (see Theorem 34.2). Suppose that

equation

Then {Mt : 0 ≤ tT} is a continuous martingale.

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

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