Questions tagged [local-martingales]

We have given number $10$ and continuous local martingale $M$ such that $\mathbb{E}|M_{\tau}|^p \le K$ for every limited stopping time $\tau$. Show that $M$ is martingale. Please help me, I have a big problem with solving this task.…

Let $X$ be a Brownian Motion with drift $\mu>0$ and $X^*$ its running maximum. Is $X^*$ a local martingale? If I construct the sequence of stopping times $\tau_k = \inf \{s >\tau_{k-1}, X(s)>X(\tau_{k-1}) \}$ and let $Y(t) = X(t) \mathbb{1} _{t \in…

$h$ is a bounded function, and $X=\{X_t;t\in [0,T]\}$ is defined by $dX_t=f(X_t)dt+\sigma(X_t) dV_t$ we defined $Z$ by $$ Z_t = \exp\left(-\int_0^th(X_s)dV_s -\frac{1}{2}\int_0^th(X_s)^2 ds \right) $$ $V$ and $W$ Brownian motions. I can show that…