Let (left(B_{t} ight)_{t geqslant 0}) be a (mathrm{BM}^{1}) and denote by (g_{t}) the last zero before time

Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and denote by \(g_{t}\) the last zero before time \(t>0\).

a) Use Theorem 11.25 to give a further proof of Lévy's arc-sine law (cf. Theorem 6.20): \(\mathbb{P}\left(g_{t})=\frac{2}{\pi} \arcsin \sqrt{\frac{s}{t}}\) for all \(s \in[0, t]\).

b) Find the conditional density \(\mathbb{P}\left(\left|B_{t}\right| \in d y \mid g_{t}=s\right)\).

c) Integrate the formula appearing in Theorem 11.25 to get

\[\sqrt{\frac{2}{\pi t}} \exp \left(-\frac{y^{2}}{2 t}\right)=\int_{0}^{t} \sqrt{\frac{2}{\pi s}} \frac{y}{\sqrt{2 \pi(t-s)^{3}}} \exp \left(-\frac{y^{2}}{2(t-s)}\right) d s\]

and give a probabilistic interpretation of this identity.

Data From 11.25 Theorem 

Transcribed Image Text:

11.25 Theorem (Lvy 1939; Chung 1976). Let (Bt)to be a BM and g, the last zero before time t > 0. Then P(Btl dy, gt = ds) = y s(-s) exp(-/-)) ds dy, s 0.