3
votes
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. ...
- 15.9k
2
votes
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 ...
- 94.9k
2
votes
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 ...
- 9,181
1
vote
Accepted
Optional sampling theorem with $L^2$ bounded martingale with stopping time
Mind it's $\{T>n\}$, not $\{T<n\}$, in the reference. The function $\tilde{\mu}(A):=E[\mathbf{1}_A|M_n|],\,A \in \mathscr{F}$ is a finite measure on $(\Omega,\mathscr{F})$, because $M_n$ is $\...
- 9,181
1
vote
Conditional Expectation of Integral over square of Brownian Motion - PDE Approach
Before going into the details of my solution, I'd like to point out how the first comment was right: your computations are wrong and indeed that expectation is $1/2(T^2-t^2)$. We can see it this way:
$...
- 724
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\...
- 160k
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