Integration - Exercises

Exercise 1

probability space. Given two random variables $X$ and $Y$ in $L^2$, the variance and covariance of $X$ and $Y$ are respectively given by

\[\text{Var}(X)=\text{Cov}(X,X)=E\left[ \left( X-E[X] \right)^2 \right] \quad \text{and}\quad\text{Cov}(X,Y)=E\left[ \left( X-E[X] \right)\left( Y-E[Y] \right) \right]\]

Let $(X_n)$ be a sequence of pairwise uncorrelated random variables in $L^2$ such that $\sup \text{Var}(X_n)<\infty$. Defining

\[S_n=\frac{1}{n} \sum_{k\leq n}\left(X_k-E[X_k]\right),\]

show that $S_n\to 0$ in probability.

Exercise 2

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{G}$ be a $\sigma$-algebra such that $\mathcal{G}\subseteq \mathcal{F}$. Let further $(A_n)$ be a sequence of pairwise disjoint elements of $\mathcal{F}$ such that $P[A_n]>0$ for every $n$. Define $\mathcal{G}=\sigma(A_n\colon n)$ the $\sigma$-algebra generated by the sequence $(A_n)$. Show that 1. for every $B \in \mathcal{F}$ it holds

$$P\left[ B|\mathcal{G} \right]:=E\left[ 1_B |\mathcal{G} \right]=\sum P\left[ B |A_n \right]1_{A_n}$$

where $P[B|A_n]:=P[B|\sigma(A_n)]=P[B\cap A_n]/P[A_n]$.

for every $X \in L^1$, it holds

\[E\left[ X|\mathcal{G} \right]=\sum \frac{E\left[ 1_{A_n}X \right]}{P[A_n]}1_{A_n}\]

Exercise 3

We consider a very simple financial market with two stocks $S^1$ and $S^2$ which values tomorrow depends on three states, that is $\Omega:=\{\omega_1,\omega_2,\omega_3\}$. The values are given as follows

\[ S^1(\omega):= \begin{cases} 90 &\text{ if }\omega=\omega_1\text{ or }\omega=\omega_2\\ 110 &\text{ if }\omega =\omega_3 \end{cases} \quad \text{and}\quad S^2(\omega):= \begin{cases} 90 &\text{ if }\omega=\omega_1\\ 100 &\text{ if }\omega=\omega_2\\ 110 &\text{ if }\omega =\omega_3 \end{cases} \]

We set $\mathcal{F}=2^\Omega$ and consider that each state comes with the same probability, that is, we consider the uniform probability measure $P$ on $\mathcal{F}$ given by $P[\{\omega_1\}]=P[\{\omega_2\}]=P[\{\omega_3\}]=1/3$. Suppose that you are an insider that have the knowledge about the outcome of $S^1$ tomorrow. Compute the conditional expected value of $S^2$ with respect to this information, that is $E[S^2|S^1]$.[^1]

Exercise 4

sequence $(X_n)$ of random variables converges to $X$ in probability if $P[|X_n-X|\geq \varepsilon]\to 0$ for every $\varepsilon>0$. Throughout the exercise $(X_n)$ and $(Y_n)$ denote sequences of random variables and $X,Y$ two random variables. 1. Show that

$$d(X,Y)=E\left[ \frac{|X-Y|}{1+|X-Y|} \right],$$

defines a metric on $L^0$ and that convergence in this metric is equivalent to convergence in probability.[^2]

Show that $X_n \to X$ $P$-almost surely implies that $X_n\to X$ in probability. Give and example that the reciprocal is not true.
Suppose that $\sum P[|X_n-X|\geq \varepsilon] <\infty$ for every $\varepsilon>0$. Show that $X_n\to X$ $P$-almost surely.
Show that each converging sequence of random variables that converges in probability has a subsequence that converges $P$-almost surely.
Suppose that any subsequence of $(X_n)$ admits itself another subsequence that converges to $X$ $P$-almost surely. Show that $X_n\to X$ in probability.
(this one is Bonus) Let $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be a continuous function.[^3] Show that if $X_n\to X$ and $Y_n\to Y$ both in probability, then it holds $f(X_n,Y_n)\to f(X,Y)$ in probability.

Exercise 5

operator is defined as[^4]

\[\|X\|_{\infty}=\inf\left\{ m\in \mathbb{R}_+\colon P\left[ |X|\geq m \right]=0 \right\}\]

for a random variable $X$. Let now $(X_n)$ be a sequence of random variables which converges $P$-almost surely to a random variable $X$. Show that for every $\varepsilon>0$, there exists a measurable set $A$ with $P[A^c]<\varepsilon$ such that

\[\lim \|(X_n-X)1_A\|_{\infty}=0.\]

Hint: Define $A_{n,k}=\cup_{m\geq n}\{|X_m-X|\geq 1/k\}$ and show that its probability can be made arbitrarily small.

Exercise 6

If $X$ and $Y$ are two independent and integrable random variables, then

\[E\left[ X \big | X+Y \right] = \frac{X+Y}{2}\]
Let $\mathcal{G}_1$ and $\mathcal{G}_2$ be two $\sigma$-algebra independent of each other. If $X$ is an integrable random variable such that $\sigma(\sigma(X),\mathcal{G}_1)$ is independent of $\mathcal{G}_2$, then

\[E\left[ X |\sigma(\mathcal{G}_1,\mathcal{G}_2)) \right]=E\left[ X|\mathcal{G}_1 \right]\]

Exercise 7

Let $X$ and $Y$ be two random variable with a joint distribution $P_{(X,Y)}$ absolute continuous with respect to the Lebesgue measure on $\mathbb{R}^2$. Show that - show that $P_{(X,Y)}$ has a density, that it, there exists a measurable function $f_{(X,Y)}:\mathbb{R}^2\to \mathbb{R}_+$, such that

$$E\left[ g(X,Y) \right]=\int_{\mathbb{R}^2} g(x,y)f_{(X,Y)}(x,y) dx dy$$

for every positive measurable function $g:\mathbb{R}^2\to [0,\infty[$.

show that $P_X$ as well as $P_Y$ are also absolutely continuous with respect to Lebegue. Provide an expression for their density $f_X$ and $f_Y$ respectively.
show that $P_{(X,Y)}=P_X\otimes K$ for some stochastic kernel

\[K(x,B)=\int_{B}^{} f_{(Y|X)}(x,y)dy\]

where $f_{(Y|X)}:\mathbb{R}^2\to \mathbb{R}$ is to be determined as a function of $f_X$, $f_Y$ and $f_{(X,Y)}$.
show that

\[E\left[ g(X,Y) | X\right]=\int_{\mathbb{R}}^{} g(X,y)f_{Y|X}(X,y) dy\]

for every positive measurable random variable $g:\mathbb{R}^2\to \mathbb{R}$.
Suppose that $f_{(X,Y)}=21_{[0,1]}(x+y)1_{[0\infty[}(x)1_{[0,\infty[}(y)$. Compute $E[\exp(Y)|X]$ explicitly.

Exercise 8

Give an example of a sequence $(X_n)$ of integrable random variable which is bounded in $L^1$ but not uniformly integrable.
Give an example of a sequence of random variables $X_n$ converging almost surely to $X$ in $L^1$ with $\sup E[|X_n|]<\infty$ but $X_n$ does not converge in $L^1$ to $X$.
For two uniformly integrable families $\mathcal{X}$ and $\mathcal{Y}$ of random variables. Show that

$$ \begin{aligned} \mathcal{Z} & := {Z\in L^0\colon |Z|\leq |X|\text{ for some }X\in \mathcal{X}}\ \mathcal{X}+\mathcal{Y} & := {X+Y\colon X\in \mathcal{X} \text{ and }Y\in \mathcal{Y}}\ \text{conv}(\mathcal{X}) & := \left{\sum \alpha k X_k \colon \text{for all } X_1,\ldots,X_n \in \mathcal{X}\text{ and }\alpha_1,\ldots \alpha_n \geq 0\text{ with }\sum^n \alpha_k=1\right}

\end{aligned} $$

are all uniformly integrable. The last set is the smallest convex set containing $\mathcal{X}$.
Construct a uniformly integrable adapted process $X=(X_t)_{t \in \mathbb{N}_0}$ on a filtrated probability space such that the family

\[\{X_\tau \colon \tau \text{ is a bounded stopping time}\}\]

is not uniformly integrable.