The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 461 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Continuous-State, Continuous-Time Models 461

The Diffusion Invariance Principle

Note that Girsanov’s Theorem requires neither that the process X be a

martingale nor that Q be an equivalent martingale measure. If X is

indeed a martingale under Q, an implication of Girsanov’s Theorem is

the diffusion invariance principle which can be stated as follows. Let X

be an Itô process:

dXt = μtdt + σtdBt

If X is a martingale with respect to an equivalent probability measure Q,

then there is a standard Brownian motion Bˆ

T in R

D under Q such that

dXt = σtdBˆt

Let’s now apply the previous results to a price process X = (V,S^1 ,...,SN–1)

where

dSt = μtdt + σtdBt

and

dVt = rtVtdt

If the short-term rate r is bounded, V–^1

t is a regular deflator. Con-

sider the deflated processes:

Zt = StVt

• 1

By Itô’s lemma, this process satisfies the following stochastic equation:

 μt  σt

dZt = –rtZt + ------ dt + ------dBt

 Vt Vt

Suppose there is an equivalent martingale measure Q. Under the

equivalent martingale measure Q, the discounted price process

Zt = StVt

• 1

is a martingale. In addition, by the diffusion invariance principle there is

a standard Brownian motion Bˆt in RD under Q such that: