HW3 MA5161. Due February 15

Exercise. (First hitting time of a closed set by a continuous stochastic process)
Let $(X_t)_{t \ge 0}$ be a continuous process adapted to a filtration $(\mathcal{F}_t)_{t \ge 0}$. Let

$T=\inf \{ t \ge 0, X_t \in F \},$

where $F$ is a closed subset of $\mathbb{R}$. Show that $T$ is a stopping time of the filtration $(\mathcal{F}_t)_{t\ge 0}$.

Exercise. (Closed martingale)
Let $(\mathcal{F}_t)_{t \ge 0}$ be a filtration defined on a probability space $(\Omega, \mathcal{F},\mathbb{P})$ and let $X$ be an integrable and $\mathcal{F}$-measurable random variable. Show that the process $\left( \mathbb{E}(X\mid \mathcal{F}_t) \right)_{t \ge 0}$ is a martingale with respect to the filtration $(\mathcal{F}_t)_{t \ge 0}$.

Exercise. Let $(\mathcal{F}_t)_{t \ge 0}$ be a filtration defined on a probability space $(\Omega, \mathcal{F},\mathbb{P})$ and let $(M_t)_{t \ge 0}$ be a submartingale with respect to the filtration $(\mathcal{F}_t)_{t \ge 0}$. Show that the function $t \rightarrow \mathbb{E} (M_t)$ is non-decreasing.

Exercise. Let $(\mathcal{F}_t)_{t \ge 0}$ be a filtration defined on a probability space $(\Omega, \mathcal{F},\mathbb{P})$ and let $(M_t)_{t \ge 0}$ be a martingale with respect to the filtration $(\mathcal{F}_t)_{t \ge 0}$. Let now $\psi : \mathbb{R} \rightarrow \mathbb{R}$ be a convex function such that for $t \ge 0$, $\mathbb{E} \left( \mid \psi(M_t) \mid \right) < + \infty$. Show that the process $(\psi(M_t))_{t \ge 0}$ is a submartingale.

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