Exer Let u be a fixed number in ℝ, and define the convex function φ(x)=e^u x for all x ∈ℝ. Let X

Step 1: Understand the given informationu000AWe are given a convex function $u005Cvarphi(x) u003D e^{ux}$, w

Question

Exer Let $u$ be a fixed number in $\mathbb{R}$, and define the convex function $\varphi(x)=e^{u x}$ for all $x \in \mathbb{R}$. Let $X$ be a normal random variable with mean $\mu=\mathbb{E} X$ and standard deviation $\sigma=\left[\mathbb{E}(X-\mu)^2\right]^{\frac{1}{2}}$, i.e., with density $$ f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}} . $$ (i) Verify that $$ \mathrm{E} e^{u X}=e^{u \mu+\frac{1}{2} u^2 \sigma^2} . $$ (ii) Verify that Jensen's inequality holds (as it must): $$ \mathbb{E} \varphi(X) \geq \varphi(\mathbb{E} X) . $$

   Exer  Let $u$ be a fixed number in $\mathbb{R}$, and define the convex function $\varphi(x)=e^{u x}$ for all $x \in \mathbb{R}$. Let $X$ be a normal random variable with mean $\mu=\mathbb{E} X$ and standard deviation $\sigma=\left[\mathbb{E}(X-\mu)^2\right]^{\frac{1}{2}}$, i.e., with density
$$
f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}} .
$$
(i) Verify that
$$
\mathrm{E} e^{u X}=e^{u \mu+\frac{1}{2} u^2 \sigma^2} .
$$
(ii) Verify that Jensen's inequality holds (as it must):
$$
\mathbb{E} \varphi(X) \geq \varphi(\mathbb{E} X) .
$$

Stochastic Calculus for Finance II : Continuous-Time Models

Steven E. Shreve 1st Edition

Chapter 1, Problem 6

Instant Answer

Step 1

We also have a normal random variable $X$ with mean $\mu = \mathbb{E}X$ and standard deviation $\sigma = \left[\mathbb{E}(X-\mu)^2\right]^{\frac{1}{2}}$. The density function of $X$ is given by: $$ f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-\mu)^2}{2 Show more…

Show all steps

Thanks for your feedback!

Exer Let $u$ be a fixed number in $\mathbb{R}$, and define the convex function $\varphi(x)=e^{u x}$ for all $x \in \mathbb{R}$. Let $X$ be a normal random variable with mean $\mu=\mathbb{E} X$ and standard deviation $\sigma=\left[\mathbb{E}(X-\mu)^2\right]^{\frac{1}{2}}$, i.e., with density $$ f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}} . $$ (i) Verify that $$ \mathrm{E} e^{u X}=e^{u \mu+\frac{1}{2} u^2 \sigma^2} . $$ (ii) Verify that Jensen's inequality holds (as it must): $$ \mathbb{E} \varphi(X) \geq \varphi(\mathbb{E} X) . $$

Play audio

Feedback

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later

Please give Ace some feedback