stochastic processes

Question

Can someone please help me with this problem?

Consider the probability space $\Omega = \{1,2,3,4,\ldots\}$ with a probability $P$ given by $P({i}) =p_i.$ Naturally, $p_i > 0$ and $\sum p_i = 1.$ The filtration $F_n$ is $\{ \{1\},\{2\},\ldots,\{n\}, \{n+1,n+2,\ldots\}\}.$ Let $X$ be a random variable defined by $X(i) = x_i.$

what does it means in this case that $E(X^2)$ is finite. Find $E(X\mid F_n).$

Thanks!

Answer 1

For $\mathbb{P} = \sum_{i = 1}^{\infty} p_i \cdot \delta_i$ (where $\delta_i$ denotes the dirac measure on $\{i\}$), $f \in L^1(\mathbb{P})$ the equality

$$\int_{\Omega} f \, d\mathbb{P} = \sum_{i=1}^{\infty} p_i \cdot f(i)$$

holds. Thus "$\mathbb{E}(X^2)$ finite" means $$\mathbb{E}(X^2) = \sum_{i=1}^{\infty} p_i \cdot X(i)^2 = \sum_{i=1}^{\infty} p_i \cdot x_i^2 < \infty$$

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Concerning ${F}_n$: Note that $(F_n)_{n \in \mathbb{N}}$ is not a filtration! Filtration means

• $F_n$ is a $\sigma$-algebra for $n \in \mathbb{N}$
• $\mathcal{F}_n \subseteq \mathcal{F}_{n+1}$ for all $n \in \mathbb{N}$

Neither is fulfilled for the given $F_n$. So probably you want to consider the $\sigma$-algebra generated by the given sets?

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To find $\mathbb{E}(X \mid F_n)$ you should use the following theorem

Let $X \in L^1$ a random variable, $\mathcal{F}$ a sub-$\sigma$-algebra and $Y \in L^1(\mathcal{F})$. Then $Y = \mathbb{E}(X \mid \mathcal{F})$ if and only if $$\forall F \in \mathcal{F}: \int_F Y \, d\mathbb{P} = \int_F X \, d\mathbb{P}$$ If $\mathcal{F} = \sigma(\mathcal{G})$, then it suffices to check the equality for $G \in \mathcal{G}$.

(If you don't get along with it, don't hesitate to ask. I'll add some more hints in this case.)