Tagged Questions

0
votes
1answer
57 views

Fail of optional sampling theorem

Could anyone help me see why the optional sampling theorem ($E(M_{\tau}\mid\mathcal{F}_{\sigma})=M_{\sigma}$ a.s.) fails for certain stopping times $\sigma\leq\tau$ for the not uniformly integrable ...
1
vote
1answer
88 views

Doob's supermartingale inequality

I'm trying to prove that For a non-negative supermartingale $M$ it holds that for all $\lambda>0$ we have $$\lambda P\{\sup_{n}M_{n}\geq\lambda\}\leq E(M_{0})$$ My idea was to use Markov's ...
3
votes
1answer
110 views

Martingale not uniformly integrable

I've come across a statement that implies that non-negative martingales for which $\{M_{\tau}\mid \tau \ \rm{stopping} \ \rm{time}\}$ is not uniformly integrable exist. I personally can't think of an ...
2
votes
1answer
37 views

Integral Martingale

I've come across some statements in measure theory that I don't fully understand. Consider the unit interval $I=[0,1]$ equipped with the Borel-$\sigma$-algebra $\mathcal{B}([0,1])$ and the Lebesgue ...
3
votes
1answer
59 views

Martingale preservation under independent enlargement of filtration

I think this is probably a very easy question but I haven't worked with $\sigma$-algebras in depth for a long time now so am finding myself a little rusty. Would be very grateful if someone could give ...
2
votes
0answers
52 views

Measure theoretic question about an inequality

Suppose I have a stoch. process $X=(X_n)$, martingale $M=(M_n)$ null at zero and constants $C\ge 0$,$\lambda>0$. I know that there exists an bounded stopping time, sucht that $E[X_\tau] \ge C$. ...
12
votes
3answers
1k views

The Laplace transform of the first hitting time of Brownian motion

Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is ...
1
vote
1answer
1k views

martingale and filtration

As I understand, martingale is a stochastic process (i.e., a sequence of random variables) such that the conditional expected value of an observation at some time $t$, given all the observations up to ...