Let (left(B_{t} ight)_{t geqslant 0}, d_{t}) and (g_{t}) be as in Corollary 11.26. Define [L_{t}^{-}:=t-g_{t} quad text

Let \(\left(B_{t}\right)_{t \geqslant 0}, d_{t}\) and \(g_{t}\) be as in Corollary 11.26. Define

\[L_{t}^{-}:=t-g_{t} \quad \text { and } \quad L_{t}:=d_{t}-g_{t}\]

Find the laws of \(\left(L_{t}^{-}, L_{t}\right), L_{t}^{-}, L_{t}\) and \(L_{t}\) conditional on \(L_{t}^{-}=r\).

Find the probability that \(\mathbb{P}\left(L_{t}>r+s \mid L_{t}^{-}=r\right)\), i.e. the probability that an excursion continues for \(s\) units of time if we know that it is already \(r\) units of time under way.

Data From Corollary 11.26

Transcribed Image Text:

11.26 Corollary. Let gt and d; the last zero before and the first zero after time t > 0. Then Ex. 11.12 P(gte ds, de du) = ds du 2s(u-s) 0