**Exercise.*** (First hitting time of a closed set by a continuous stochastic process)
Let be a continuous process adapted to a filtration . Let*

*where is a closed subset of . Show that is a stopping time of the filtration .*

**Exercise. (Closed martingale)**

*Let be a filtration defined on a probability space and let be an integrable and -measurable random variable. Show that the process is a martingale with respect to the filtration .
*

**Exercise.** *Let be a filtration defined on a probability space and let be a submartingale with respect to the filtration . Show that the function is non-decreasing.
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**Exercise.** *Let be a filtration defined on a probability space and let be a martingale with respect to the filtration . Let now be a convex function such that for , . Show that the process is a submartingale.*