Let (Wt)t≥0(W_t)_{t \geq 0}(Wt)t≥0 be a standard Brownian motion and define the hitting time of level a∈Ra \in \mathbb{R}a∈R by:
Questions:
compute for a∈Ra \in \mathbb{R}a∈R and u∈R+u \in \mathbb{R}^+u∈R+: Mτa(u)M_{\tau_a}(u)Mτ
Show that τa<+∞\tau_a < +\inftyτa<+∞ almost surely, and that E[τa]=+∞\mathbb{E}[\tau_a] = +\inftyE[τa]=+∞.