. In Example 1.6.6, we began with a standard normal random...

. In Example 1.6.6, we began with a standard normal random variable $X$ under a measure $\boldsymbol{P}$. According to Exercise 1.6, this random variable has the moment-generating function $$ \mathbb{E} e^{u X}=e^{\frac{1}{2} u^2} \text { for all } u \in \mathbb{R} . $$ The moment-generating function of a random variable determines its distribution. In particular, any random variable that has moment-generating function $e^{\frac{1}{2} u^2}$ must be standard normal. In Example 1.6.6, we also defined $Y=X+\theta$, where $\theta$ is a constant, we set $Z=e^{-\theta X-\frac{1}{2} \theta^2}$, and we defined $\widetilde{\mathbb{P}}$ by the formula (1.6.9): $$ \widetilde{\mathbb{P}}(A)=\int_A Z(\omega) d \mathbb{P}(\omega) \text { for all } A \in \mathcal{F} . $$ We showed by considering its cumulative distribution function that $Y$ is a standard normal random variable under $\widetilde{\mathbb{P}}$. Give another proof that $Y$ is standard normal under $\widetilde{\mathbb{P}}$ by verifying the moment-generating function formula $$ \widetilde{\mathbb{E}} e^{u Y}=e^{\frac{1}{2} u^2} \text { for all } u \in \mathbb{R} \text {. } $$

. In Example 1.6.6, we began with a standard normal random variable $X$ under a measure $\boldsymbol{P}$. According to Exercise 1.6, this random variable has the moment-generating function
$$
\mathbb{E} e^{u X}=e^{\frac{1}{2} u^2} \text { for all } u \in \mathbb{R} .
$$

The moment-generating function of a random variable determines its distribution. In particular, any random variable that has moment-generating function $e^{\frac{1}{2} u^2}$ must be standard normal.

In Example 1.6.6, we also defined $Y=X+\theta$, where $\theta$ is a constant, we set $Z=e^{-\theta X-\frac{1}{2} \theta^2}$, and we defined $\widetilde{\mathbb{P}}$ by the formula (1.6.9):
$$
\widetilde{\mathbb{P}}(A)=\int_A Z(\omega) d \mathbb{P}(\omega) \text { for all } A \in \mathcal{F} .
$$

We showed by considering its cumulative distribution function that $Y$ is a standard normal random variable under $\widetilde{\mathbb{P}}$. Give another proof that $Y$ is standard normal under $\widetilde{\mathbb{P}}$ by verifying the moment-generating function formula
$$
\widetilde{\mathbb{E}} e^{u Y}=e^{\frac{1}{2} u^2} \text { for all } u \in \mathbb{R} \text {. }
$$

Key Concepts

Exponential Tilting

Exponential tilting, also known as the Esscher transform in some contexts, is a method used to modify the probability measure by reweighting events with an exponential factor. This technique is used to shift the mean or change the distribution of a random variable, and it plays a key role in ensuring that after the change of measure, the random variable of interest (after a linear shift) still possesses the distributional characteristics, such as those of the standard normal distribution, as verified by its moment generating function.

Radon-Nikodym Derivative

The Radon-Nikodym derivative is a function that serves as the density of one measure with respect to another when the first measure is absolutely continuous with respect to the second. It is a crucial component when performing a change of measure, as it allows integration with respect to the new measure to be expressed in terms of the original measure. This concept is central in proving that a transformation of a random variable results in a certain distribution under the new measure.

Standard Normal Distribution

The standard normal distribution is a fundamental probability distribution characterized by a mean of zero and a variance of one. It has a specific moment generating function given by exp(u^2 / 2) for all real numbers u, which uniquely identifies the distribution. This property is often used to verify that a transformed or shifted random variable retains the characteristics of a standard normal variable under an appropriate measure.

Moment Generating Function

The moment generating function (MGF) of a random variable is a tool used to uniquely characterize its probability distribution. It is defined as the expected value of the exponential function of the random variable scaled by a parameter. The MGF can be used to derive moments of the distribution and, importantly, if two random variables have the same MGF in an open neighborhood around zero, then they have the same distribution.

Change of Measure

Change of measure is a technique in probability theory that involves switching from one probability measure to another using a Radon-Nikodym derivative. This is particularly useful in areas like financial mathematics and stochastic calculus, where one may wish to transform a model under a risk-neutral measure or another equivalent measure. It allows properties of random variables to be analyzed under different probabilistic frameworks while maintaining absolute continuity between the measures.

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