The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 460 Wednesday, February 4, 2004 1:08 PM

460 The Mathematics of Financial Modeling and Investment Management

t t

Xt = x + ∫μs sd + ∫σsdBs

0 0

In this process, μs is an N-vector process and σs is an N × D matrix.

Suppose that there are both a vector process ν = (ν^1 ,...,νN) and a vector

process θ = (θ^1 ,...,θN) such that σtθt = μt – νt where the product σtθt is

not a scalar product but is performed component by component. Sup-

pose, in addition, that the process θ satisfies the Novikov condition:

 1

∫ 0

t ⋅

--- θθsd



2 

Ee ∞<

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

lowing integral

t

Bˆ

t = Bt + ∫θs sd

0

defines a standard Brownian motion Bˆt in RD 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 essentially states that under technical condi-

tions (the Novikov condition) by changing the probability measure, it is

possible to transform an Itô process into another Itô process with arbi-

trary drift. Prima facie, this result might seem unreasonable. In the end

the drift of a process seems to be a fundamental feature of the process as

it defines, for example, the average of the process. Consider, however,

that a stochastic process can be thought as the set of all its possible

paths. In the case of an Itô process, we can identify the process with the

set of all continuous and square integrable functions. As observed

above, the drift is an average and it is determined by the probability

measure on which the process is defined. Therefore, it should not be sur-

prising that by changing the probability measure it is possible to change

the drift.