conditional expectation under change of
measure∗
stevecheng†
2013-03-21 22:29:34
Let P be a given probability measure on some σ-algebra F. Suppose a
new probability measure Q is defined by dQ = Z dP, using some F-measurable
random variable Z as the Radon-Nikodym derivative. (Necessarily we must
have Z ≥ 0 almost surely, and EZ = 1.)
We denote with E the expectation with respect to the measure P, and with
EQ the expectation with respect to the measure Q.
Theorem 1. If Q is restricted to a sub-σ-algebra G ⊆ F, then the restriction has
the conditional expectation E[Z | G] as its Radon-Nikodym derivative: dQ|G =
E[Z | G] dP|G .
In other words,
dQ|G
dQ
=
.
dP|G
dP |G
Proof. It is required to prove that, for all B ∈ G,
Q(B) = E E[Z | G] 1B .
But this follows at once from the law of iterated conditional expectations:
E E[Z | G] 1B = E E[Z1B | G] = E[Z1B ] = Q(B) .
Theorem 2. Let G ⊆ F be any sub-σ-algebra. For any F-measurable random
variable X,
E[Z | G] EQ [X | G] = E[ZX | G] .
That is,
dQ
dP
dQ
E [X | G] = E
X|G .
dP
Q
|G
∗ hConditionalExpectationUnderChangeOfMeasurei
hstevechengi version: h39165i Privacy setting: h1i
created:
h2013-03-21i
by:
hDerivationi h60A10i h60-00i
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Proof. Let Y = E[Z | G], and B ∈ G. We find:
EQ 1B E[ZX | G] = E Y 1B E[ZX | G]
= E E[Y 1B ZX | G]
(since dQ|G = Y dP|G )
= E[Y 1B ZX]
= EQ [Y 1B X]
= EQ 1B EQ [Y X | G] .
(since dQ = Z dP)
Since B ∈ G is arbitrary, we can equate the G-measurable integrands:
E[ZX | G] = EQ [Y X | G] = Y EQ [X | G] .
Observe that if dQ/dP > 0 almost surely, then
.
dQ
dQ
EQ [X | G] = E
X|G
.
dP
dP |G
Theorem 3. If Xt is a martingale with respect to Q and some filtration {Ft },
then Xt Zt is a martingale with respect to P and {Ft }, where Zt = E[Z | Ft ].
Proof. First observe that Xt Zt is indeed Ft -measurable. Then, we can apply
Theorem 2, with X in the statement of that theorem replaced by Xt , Z replaced
by Zt , F replaced by Ft , and G replaced by Fs (s ≤ t), to obtain:
E[Xt Zt | Fs ] = Zs EQ [Xt | Fs ] = Zs Xs ,
thus proving that Xt Zt is a martingale under P and {Ft }.
Sometimes the random variables Zt in Theorem 3 are written as dQ
dP t . (This
is a Radon-Nikodym derivative process; note that Zt defined as Zt = E[Z | Ft ]
is always a martingale under P and {Ft }.)
Under the hypothesis Zt > 0, there is an alternate restatement of Theorem
3 that may be more easily remembered:
Theorem 4. Let Zt = (dQ/dP)t > 0 almost surely. Then Xt is a martingale
with respect to P, if and only if Xt /Zt is a martingale with respect to Q.
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