# Tagged Questions

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### How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
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### Do we need $\tau \leq \nu$ to show $E(X_\tau)=E(X_\nu)$?

My lecture notes claim that if $(X_n)$ is a martingale and $\tau$ is a stopping time bounded by $N$ then $$E(X_\tau)=E(X_{\tau \wedge N})=E(X_{\tau \wedge 0})=E(X_0)$$ and then remarks that if $\tau$ ...
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### Prove hitting time (with $\tau_a:=n+1$ if $S_k\leq a \forall 0\leq k\leq n$)

I have random variable $\tau$ defined as follows $\tau_a=\min\{0 \leq k \leq n : S_k>a\}$ $(a>0)$ $\tau_a:=n+1$ if $S_k\leq a$, $\forall 0\leq k\leq n$ (how should I care during the proof?) ...
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### Optional Sampling Theorem in discrete setting

I have a question about proving the optional sampling theorem in discrete setting. I dont know if what I am doing is mathematical justified. Can someone help me with this? Defenition: Let $\tau$ be a ...
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### Stopping times, Filtration, Martingales,

I am new here and I have a question. Defenition: Let $\tau$ be a stopping time, then $\digamma_{\tau}=\{F\subset \Omega: \forall n \in N \cup \{\infty\} , F\cap(\tau\leq n)\in \digamma_{n}$} is a ...
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### Example of a martingale and a stopping time with $E(T)<\infty$ but $E(X_T) \neq E(X_0)$

Is there an example of a martingale in discrete time $X_0, X_1, X_2,\ldots$ and a stopping time $T$ so that $E(T) <\infty$ but $E(X_T) \neq E(X_0)$? With added assumptions on how $X_n$ behaves, ...
### $(S_n^2-n)_{n\ge 0}$ martingale and bounded stopping times
Consider the random walk $$S_n=\sum_{k}^{n}X_{k}$$ Where $X_k$'s are iid, $$\mathbb P(X_1=1)=\mathbb P(X_1=-1)=\frac{1}{2}$$ and $\mathcal{F}_{n}=\sigma(X_i,0\leq i\leq n)$. How do I prove that ...