Computing the conditional expectation $E[X_{n+1}\mid \mathcal{F}_n]$.

Question

Let $\{X_n\}_{n\in\mathbb{N}}$ be a stochastic process consisting of the uniform random variables $X_0\sim\mathcal{U}[0,T]$, $X_1\sim\mathcal{U}[0,X_0]$ and in general $X_{n+1}\sim\mathcal{U}[0,X_n]$ for some $T>0$. Let $\{\mathcal{F}_n\}$ be it's natural filtration. How do we compute $$E[X_{n+1}\mid \mathcal{F}_n]?$$

Is the process a super-Martingale?

Answer 1

Let $U_0,U_1,U_2,\ldots$ be i.i.d $\mathcal{U}[0,1]$. Then we have $X_{n+1} = U_{n+1}X_n$. So, $ \mathbb{E}[X_{n+1}|\mathcal{F}_n] = \mathbb{E}[U_{n+1}X_{n}|\mathcal{F}_n] = X_n\mathbb{E}[U_{n+1}] = \dfrac{1}{2}X_n \leq X_n$. So the process is a super-martingale.