2 votes
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central limit theorem for continuous-time martingales

What you usually do in this kind of contexts is to deduce the continuous-time martingale theorem from the discrete-time martingale analogue theorem. There is a paper showing that kind of method : ...
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2 votes
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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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  • 9,241
1 vote
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Gambler's Ruin: Expected time of ruin using martingale and stopping time

Regarding the ruin time $T_0$: Apply optional stopping to the martingale $M_n=X_n-\mu n=X_n+n/5$ at the bounded stopping time $T_0 \wedge n=\min(T_0,n)$: $$x=E(M_0)=E(M_{T_0 \wedge n})=E(X_{T_0 \wedge ...
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