Let \(\tau:=\tau_{(-a, b)^{c}}^{\circ}\) be the first entrance time of a \(\mathrm{BM}^{1}\) into the set \((-a, b)^{c}\).

a) Show that \(\tau\) has finite moments \(\mathbb{E} \tau^{n}\) of any order \(n \geqslant 1\).

Use Example 5.2.d) and show that \(\mathbb{E} e^{c \tau}0\).

b) Evaluate \(\mathbb{E} \int_{0}^{\tau} B_{s} d s\).

Find a martingale which contains \(\int_{0}^{t} B_{s} d s\).

Data From 5.2.d Example 

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5.2 Example. Let (B+)120, Bt = (B,..., B), be a d-dimensional Brownian motion and ()to an admissible filtration. a) (B)tzo is a martingale with respect to Ft.