conditional expectation under change of measure

Let be a given probability measureMathworldPlanetmath on some σ-algebra . Suppose a new probability measure is defined by d=Zd, using some -measurable random variableMathworldPlanetmath Z as the Radon-Nikodym derivativeMathworldPlanetmath. (Necessarily we must have Z0 almost surely, and 𝔼Z=1.)

We denote with 𝔼 the expectation with respect to the measure , and with 𝔼 the expectation with respect to the measure .

Theorem 1.

If Q is restricted to a sub-σ-algebra GF, then the restrictionPlanetmathPlanetmath has the conditional expectation E[ZG] as its Radon-Nikodym derivative: dQG=E[ZG]dPG.

In other words,


It is required to prove that, for all B𝒢,

(B)=𝔼[𝔼[Z𝒢] 1B].

But this follows at once from the law of iterated conditional expectations:

𝔼[𝔼[Z𝒢] 1B]=𝔼[𝔼[Z1B𝒢]]=𝔼[Z1B]=(B).
Theorem 2.

Let GF be any sub-σ-algebra. For any F-measurable random variable X,


That is,


Let Y=𝔼[Z𝒢], and B𝒢. We find:

𝔼[1B𝔼[ZX𝒢]] =𝔼[Y1B𝔼[ZX𝒢]] (since d𝒢=Yd𝒢)
=𝔼[Y1BX] (since d=Zd)

Since B𝒢 is arbitrary, we can equate the 𝒢-measurable integrands:


Observe that if d/d>0 almost surely, then

Theorem 3.

If Xt is a martingaleMathworldPlanetmath with respect to Q and some filtrationPlanetmathPlanetmath {Ft}, then XtZt is a martingale with respect to P and {Ft}, where Zt=E[ZFt].


First observe that XtZt is indeed t-measurable. Then, we can apply Theorem 2, with X in the statement of that theorem replaced by Xt, Z replaced by Zt, replaced by t, and 𝒢 replaced by s (st), to obtain:


thus proving that XtZt is a martingale under and {t}. ∎

Sometimes the random variables Zt in Theorem 3 are written as (dd)t. (This is a Radon-Nikodym derivative process; note that Zt defined as Zt=𝔼[Zt] is always a martingale under and {t}.)

Under the hypothesisMathworldPlanetmath 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.

Title conditional expectation under change of measure
Canonical name ConditionalExpectationUnderChangeOfMeasure
Date of creation 2013-03-22 16:54:21
Last modified on 2013-03-22 16:54:21
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 9
Author stevecheng (10074)
Entry type Derivation
Classification msc 60A10
Classification msc 60-00
Related topic Martingale
Related topic ConditionalExpectation