The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 459 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Continuous-State, Continuous-Time Models 459

EQUIVALENT MARTINGALE MEASURES AND

GIRSANOV’S THEOREM

We first need to establish an important mathematical result known as

Girsanov’s Theorem. This theorem applies to Itô processes. Let’s first

state Girsanov’s theorem in simple cases. Let X be a single-valued Itô

process where B is a single-valued standard Brownian motion:

t t

Xt = x + ∫μs sd + ∫σsdBs

0 0

Suppose that a process ν and a process θ such that σtθt = μt – νt are

given. Suppose, in addition, that the process θ satisfies the Novikov con-

dition which requires

 1 t^2

--- θ sd



2 ∫^0 s 

Ee ∞<

Then, there is a probability measure Q equivalent to P such that the fol-

lowing integral

t

Bˆt = B

t + ∫θs sd

0

defines a standard Brownian motion Bˆ

t in R on (Ω,ℑ,Q) with the same

standard filtration of the original Brownian motion Bt. In addition,

under Q the process X becomes

t t

Xt = x + ˆ

∫νs sd + ∫σs dBs

0 0

Girsanov’s Theorem states that we can add drift to a standard

Brownian motion and still obtain a standard Brownian motion under

another probability measure. In addition, by changing the probability

measure we can arbitrarily change the drift of an Itô process.

The same theorem can be stated in multiple dimensions. Let X be an

N-valued Itô process: