Let (tau=inf left{t: M_{t}-W_{t}>a ight}). Prove that (M_{tau}) follows the exponential law with parameter (a^{-1}). The exponential
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Let \(\tau=\inf \left\{t: M_{t}-W_{t}>a\right\}\). Prove that \(M_{\tau}\) follows the exponential law with parameter \(a^{-1}\).
The exponential law stems from
\[\mathbb{P}\left(M_{\tau}>x+y \mid M_{\tau}>y\right)=\mathbb{P}\left(\tau>T_{x+y} \mid \tau>T_{y}\right)=\mathbb{P}\left(M_{\tau}>x\right)\]
The value of the mean of \(M_{\tau}\) is obtained by passing to the limit in the equality \(\mathbb{E}\left(M_{\tau \wedge n}\right)=\mathbb{E}\left(M_{\tau \wedge n}-W_{\tau \wedge n}\right)\).
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Mathematical Methods For Financial Markets
ISBN: 9781447125242
1st Edition
Authors: Monique Jeanblanc, Marc Yor, Marc Chesney
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