Strict Local Martingale. Here, we give an example of a local martingale which is not a martingale, i.e., a strict local martingale. Let \(M\) be a continuous martingale such that \(M_{0}=1\) and define \(T_{0}=\inf \left\{t: M_{t}=0\right\}\). We assume that \(\mathbb{P}\left(T_{0}<\infty\right)=1\). We introduce the probability measure \(\mathbb{Q}\) as \(\left.\mathbb{Q}\right|_{\mathcal{F}_{t}}=\left.M_{t \wedge T_{0}} \mathbb{P}\right|_{\mathcal{F}_{t}}\). It follows that \[\begin{equation*}
\mathbb{Q}\left(T_{0}\end{equation*}\]
i.e., \(\mathbb{Q}\left(T_{0}=\infty\right)=1\). The process \(X\) defined by \(\left(X_{t}=M_{t}^{-1}, t \geq 0\right)\) is a \(\mathbb{Q}\) local martingale and is positive. It is not a martingale: indeed its expectation is not constant \[\mathbb{E}_{\mathbb{Q}}\left(X_{t}\right)=\mathbb{E}_{\mathbb{P}}\left(\frac{M_{t \wedge T_{0}}}{M_{t}}\right)=\mathbb{P}\left(tFrom Girsanov's theorem, the process \(\widetilde{M}_{t}=M_{t}-\int_{0}^{t} \frac{d\langle Mangle_{s}}{M_{s}}\) is a \(\mathbb{Q}\)-local martingale. In the case \(M_{t}=B_{t}\), we get \[B_{t}=\beta_{t}+\int_{0}^{t} \frac{d s}{B_{s}}\]

where \(\beta\) is a \(\mathbb{Q}\)-Brownian motion. Hence, the process \(B\) is a \(\mathbb{Q}\)-Bessel process of dimension 3.