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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