# Tag Info

Accepted

### Martingale processes

Ito's formula is a wonderful tool, but it is overkill for this question. If $\alpha<0$, then $Y_t$ is undefined with positive probability for each $t>0$. If $\alpha>0$, then $Y_t>0$ a.s. ...

### A stopping time problem for a random walk with transition probabilities dependent on states

You can introduce stopping times $T_N=\inf \{ t : X_t \in \{ 0,N \} \}$. Then using optional stopping you have $(n+1)^2=P(X_{T_N}=0)+(N+1)^2 P(X_{T_N}=N) + E[T_N]$. Now you can proceed in one of two ...
Accepted

### Two questions about conditional expectations

(1). The most general formulation of the property I am aware of states that over a $\sigma$-finite measure space $(X,\mathscr{A},\mu)$, if $\mathscr{G}\subseteq \mathscr{A}$ is a sub-$\sigma$-algebra ...
1 vote
Accepted

1 vote

### Are there any positive uniformly integrable martingales which limit to 0?

You indeed solved correctly this question. Actually, this question is not bad as it shows how crucial the uniform integrability assumption is. Such martingale have the representation \$X_n=\mathbb E\...

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