Here, we give an example of a local martingale which is not a martingale, i e , a strict local martingale Let M M be a continuous martingale such that M 0 1 M 0 1 and define T 0 inf t M t 0 T 0 inf t M t 0 We assume that P ( T 0 ) 1 P ( T 0 ) 1 We introduce the probability measure Q Q as Q F t M t ...

SOLUTION Here are the key points about this example 1 Mt is a continuous martingale with M0 1 ...

Here, we give an example of a local martingale which is not a martingale, i.e., a strict

Question:

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 $T_{0} = inf {t : M_{t} = 0}$ . We assume that $P (T_{0} < \infty) = 1$ . We introduce the probability measure $Q$ as ${Q |}_{F_{t}} = {M_{t \land T_{0}} P |}_{F_{t}}$ . It follows that

$\begin{matrix} (6.1.4) & Q (T_{0} < t) = E_{P} (1_{T_{0} < t} M_{t \land T_{0}}) = 0, \end{matrix}$

i.e., $Q (T_{0} = \infty) = 1$ . The process $X$ defined by $(X_{t} = M_{t}^{- 1}, t \geq 0)$ is a $Q$ local martingale and is positive. It is not a martingale: indeed its expectation is not constant

$E_{Q} (X_{t}) = E_{P} (\frac{M_{t \land T_{0}}}{M_{t}}) = P (t < T_{0}) \neq 1 = X_{0}$

From Girsanov's theorem, the process ${\tilde{M}}_{t} = M_{t} - \int_{0}^{t} \frac{d ⟨ M ⟩_{s}}{M_{s}}$ is a $Q$ -local martingale. In the case $M_{t} = B_{t}$ , we get

$B_{t} = β_{t} + \int_{0}^{t} \frac{d s}{B_{s}}$

where $β$ is a $Q$ -Brownian motion. Hence, the process $B$ is a $Q$