Give another proof that (lim _{t ightarrow infty} M_{t}=0) in the above Example 1.2.1.7 by using

Give another proof that \(\lim _{t \rightarrow \infty} M_{t}=0\) in the above Example 1.2.1.7 by using \(T_{-a}=\inf \left\{t: W_{t}=-a\right\}\).

Example 1.2.1.7:

The martingale \(M_{t}=\exp \left(\lambda W_{t}-\frac{\lambda^{2}}{2} t\right)\) where \(W\) is a Brownian motion is \(L^{1}\) bounded (indeed \(\forall t, \mathbb{E}\left(M_{t}\right)=1\) ). From \(\lim _{t \rightarrow \infty} \frac{W_{t}}{t}=0\), a.s., we get that

\[\lim _{t \rightarrow \infty} M_{t}=\lim _{t \rightarrow \infty} \exp \left(t\left(\lambda \frac{W_{t}}{t}-\frac{\lambda^{2}}{2}\right)\right)=\lim _{t \rightarrow \infty} \exp \left(-t \frac{\lambda^{2}}{2}\right)=0\]

hence this martingale is not u.i. on \(\left[0, \infty\left[\right.\right.\) (if it were, it would imply that \(M_{t}\) is null!).