Chapter 9
Girsanov Formula
If α is Gaussian with mean b1 and variance a while β has the same variance but
a mean b2 the Radon-Nikodym derivative can be explicitly calculated
(x−b1 )2
(b2 −b1 )(x−b1 )
(b −b )2
(x−b2 )2
dβ
− 2 2a 1
a
(x) = e− 2a + 2a = e
dα
This suggests that if P ∈ I(a, b) and Q ∈ I(a, b + ac) for some bounded c, then
Z t
Z
1 t
dQ
c(s, x(s))dy(s) −
|Ft = exp[
ha(s, x(s))c(s, x(s)), c(s, x(s))ids]
dP
2 s0
s0
where
y(t) = x(t) −
Z
t
b(s, x(s))ds
s0
Theorem 9.1. With
Z t
Z
1 t
c(s, x(s))dy(s) −
R(t, ω) = exp[
ha(s, x(s))c(s, x(s)), c(s, x(s))ids]
2 s0
s0
if P ∈ I(a, b) then Q with dQ
dP |Ft = R(t, ω) is in I(a, b + ac) and conversely if
1
dP
Q ∈ I(a, b + ac) then P with dQ
|Ft = R(t,ω)
is in I(a, b).
Proof. If P ∈ I(a, b) then with
y(t) = x(t) −
Z
t
b(s, x(s))ds
s0
Z t
Z
1 t
c(s, x(s)) · dy(s) −
R(t, ω) = exp[
ha(s, x(s))c(s, x(s)), c(s, (x(s))ids]
2 s0
s0
1
2
CHAPTER 9. GIRSANOV FORMULA
is martingale. We can define Q by
c(s, x) + θ and will have
dQ
dP |Ft
R(t, θ, ω) = exp
Z
= R(t, ω). We can replace c by
t
(θ + c(s, x(s))) · dy(s)
s0
1
−
2
Z
t
ha(s, x(s))(θ + c(s, x(s))), (θ + c(s, (x(s)))ids
s0
= R(t, ω) exp hθ, y(t) − y(s0 )i −
1
2
−
Z
t
s0
Z
t
ha(s, x(s))c(s, x(s)), θi
s0
ha(s, (x(s))θ, θi
is a martingale for all θ. It is easy to see that this equivalent to
Z t
hθ, a(s, x(s))c(s, x(s))ids
exp hθ,y(t) − y(s0 )i −
s0
Z
1 t
ha(s, (x(s))θ, θids
−
2 s0
Z t
hθ, b(s, x(s)) + a(s, x(s))c(s, x(s))ids
= exp hθ, x(t) − x(s0 ) −
s0
−
1
2
Z
t
s0
ha(s, (x(s))θ, θids
being a martingale with respect to (C[s0 , T ], Ft, Q) i.e. Q ∈ I(a, b + ac). The
steps can be retraced to prove the converse.