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. ...
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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 ...
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  • 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 ...
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  • 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 $\...
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  • 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: $...
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  • 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\...
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