# Tagged Questions

For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.

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### Expectation of martingales [closed]

If I know that $\{M_t\}_t$ is a martingale, we know that $$\mathbb{E}({M_tM_s})=\mathbb{E}(M_{\min({t,s})}^2)$$ Is there something I can say about $\mathbb{E}(M_tM_sM_r)$?
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### Attempting to show $P(|S_n| <1)$ for a martingale $(S_n)$

Now, I am stuck on the last part of the question. I managed to find the solutions, but I don't udnerstand them completely. What I don't understand is: How they got that indicator function, and why ...
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### Show $X_t$ is a martingale, if $(M_k)_{k=0}^n$ is a discrete-time martingale, and $X_k$ = $M_k$ − $M_{k−1}$ for (k = 1, . . . , n).

If $X_1, \dots, X_n$ are independent random variables, then using the equation $$\operatorname{Var}\biggl(\sum_{i=1}^\infty X_i\biggr) = \sum_{i=1}^n \operatorname{Var}(X_i)$$ Show that this is also ...
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### Is the Stock Prices in a Perfect Market martingale or not?

Stock Prices in a Perfect Market Let Xn,, be the closing price at the end of day n of a certain publicly traded security such as a share of stock. While daily prices may fluctuate, many scholars ...
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### Almost sure and $\mathcal{L}^1$ convergence of $Y_n=(\cos a)^{-n}\cos(a(Z_1+\cdots+Z_n))$ with $(Z_n)$ i.i.d. Bernoulli

Let $(X_i)$ be i.i.d. with $P(X_i= a)=P(X_i = -a)=\frac{1}{2}$, for some $a$ such that $2a \notin\pi\mathbb Z$. Let $$Y_n=\frac1{\cos^n(a)}\cos\left(\sum_{i=1}^n X_i\right).$$ Check whether $(Y_n)$ ...
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### Expected value of the infinite product of indepedent random variables

We assume that $Y_{n}$ are independent random variables and we let $Y_{n}$ have the values $\frac{3}{2}$or $\frac{1}{2}$ with probability $\frac{1}{2}$ each. We let $X_{n}=Y_{1}\cdot \cdot \cdot Y_{n}$...
I start with a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t),P)$. I assume that the filtration is right-continuous. On this probability space I define a supermartingale $M$. Now ...