1.6 Change of Measure 1

Introduction We used a positive random variable Z to change probability measures on a space Ω. is risk-neutral probability measure P(ω) is the actual probability measure When Ω is uncountably infinite and then it will division by zero. We could rewrite this equation as But the equation tells us nothing about the relationship among and Z. Because, the value of Z(ω) However, we should do this set-by-set, rather than ω-by-ω 2

Theorem Let (Ω, F, P) be a probability space and let Z be an almost surely nonnegative random variable with EZ=1. For A F define Then is a probability measure. Furthermore, if X is a nonnegative random variable, then If Z is almost surely strictly positive, we also have for every nonnegative random variable Y. 3

Concept of Theorem

Proof of Theorem (1) According to Definition 1.1.2, to check that is a probability measure, we must verify that and that is countably additive. We have by assumption For countable additivity, let A 1, A 2,… be a sequence of disjoint sets in F, and define,. Because and, we may use the Monotone Convergence Theorem, Theorem 1.4.5, to write 5

Proof of Theorem (2) But, and so Now support X is a nonnegative random variable. If X is an indicator function X=I A, then which is When Z>0 almost surely, is defined and we may replace X in by to obtain 6

Definition ( 測度等價 ) Let Ω be a nonempty set and F a σ-algebra of subsets of Ω. Two probability measures P and on (Ω, F) are said to be equivalent if they agree which sets in F have probability zero. 7

Definition Description Under the assumptions of Theorem 1.6.1, including the assumption that Z>0 almost surely, P and are equivalent. Support is given and P(A)=0. Then the random variable I A Z is P almost surely zero, which implies On the other hand, suppose satisfies. Then almost surely under, so Equation (1.6.5) implies P(B)=EI B =0. This shows that P and agree which sets have probability zero. 8

Example (1) Let Ω=[0,1], P is the uniform (i.e., Lebesgue) measure, and Use the fact that dP(ω)=dω, then B[0,1] is a σ-algebra generated by the close intervals. Since [a,b] [0,1] implies This is with Z(ω)=2ω 9

Example (2) Note that Z(ω)=2ω is strictly positive almost surely (P{0}=0), and According to Theorem 1.6.1, for every nonnegative random variable X(ω), we have the equation This suggests the notation 10

Example Description In general, when P,, and Z are related as in Theorem 1.6.1, we may rewrite the two equations and as A good way to remember these equations is to formally write Equation is a special case of this notation that captures the idea behind the nonsensical equation that Z is somehow a “ratio of probabilities.” In Example 1.6.4, Z(ω) is in fact a ration of densities: 11

Definition (Radon-Nikodým derivative) Let (Ω, F, P) be a probability space, let be another probability measure on (Ω, F) that is equivalent to P, and let Z be an almost surely positive random variable that relates P and via (1.6.3). Then Z is called the Radon-Nikodým derivative of with respect to P, and we write 12

Example (1) (Change of measure for a normal random variable) X~N(0,1) with respect to P, Y=X+θ~N(θ,1) with respect to P. Find such that Y~N(0,1) with respect to Sol Find Z>0, EZ=1, Let, and Z>0 is obvious because Z is defined as an exponential. And EZ=1 follows from the integration 13

Example (2) Where we have made the change of dummy variable y=x+θ in the last step. But, being the integral of the standard normal density, is equal to one. where y=x+θ. It shows that Y is a standard normal random variable under the probability. 當機率分配函數定義下來之後，一切特性都定義下來了。 14

Theorem (Radon-Nikodým) Let P and be equivalent probability measures defined on (Ω, F). Then there exists an almost surely positive random variable Z such that EZ=1 and 15