Let (left(B_{t}ight)_{t geqslant 0}) be a (mathrm{BM}^{1}) and set (M_{t}=sup _{s leqslant t} B_{s}). Denote by (xi_{t})

Question:

Let $\left(B_{t}ight)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and set $M_{t}=\sup _{s \leqslant t} B_{s}$. Denote by $\xi_{t}$ the largest zero of $B_{s}$ before time $t$ and by $\eta_{t}$ the largest zero of $Y_{S}=M_{s}-B_{s}$ before time $t$. Show that $\xi_{t} \sim \eta_{t}$.

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