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Questions tagged [martingales]

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

In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Expressing the following mathematically:

A sequence of random variables $X_0, X_1, \dots$ with finite means such that the conditional expectation of $X_{n+1}$ given $X_0, X_1, X_2, \dots, X_n$ is equal to $X_n$, i.e., $$\mathbb{E}[X_{n+1}\mid X_0, X_1, \dots, X_n] = X_n.$$

A one-dimensional random walk with steps equally likely in either direction $(p=q=\frac12)$ is an example of a martingale.

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sum of submartingales is not submartingale?

Please give an example satisfying: submartingale $(X_n)\; w.r.t\; \mathcal{(F_n)}, (Y_n)\; w.r.t \;\mathcal{(G_n)}$. But $(X_n + Y_n)$ is not a submartingale $w.r.t$ any filtration. Thanks
Mike
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Finding a Doob decomposition

Let $(M_n)_{n\in \mathbb{N}}$ be a square integrable martingale and $(X_k)_{k\in \mathbb{N}}$ iid square integrable random variables with $\mathbb{E}X_1=1$. Show $M_n := \Pi_{k=1}^n X_k$ is a square integrable martingale and determine its quadratic…
Uhmm
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Can $(X_n)_{n \ge 0}$ and $(X_n^2)_{n \ge 0}$ be both martingales?

Assume we have a martingale $((X_n)_{n\ge 0}, (\mathcal{F}_n)_{n\ge 0})$, for which each random variable $X_n$ is bounded (so there is no problem with integrability) and not constant. Is it true that $X_n^2$ cannot be a martingale? My idea is that…
Barabara
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Star Trek Williams book

This is E10.10 From Williams "Probability with Martingales" book, which I spent a lot but I could not figure out what is the relation between Gauss's theorem and this end! Here is the question: The control system on the star-ship Enterprise has gone…
Pizzaro
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If $X_n$ is a martingale, is $X_n^2$ a submartingale?

Let $(X_n,F_n)$ (n = 1,2,...) be a martingale. Is it true that $X_n^2$ is a submartingale?
Kerry
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Martingales and Integrals question

I'm stuck with an martingales exercise here: $$\lim_{n\to\infty}\int_0^1\int_0^1\cdots\int_0^1\sin\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)dx_1dx_2\cdots dx_n$$ I tried to do it without martingales and it work fine for me. But I can't see how to…
Eva Leon
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Martingale and Submartingale problem

Let $T_{1},T_{2},\ldots$ be an iid sequence with distribution function $F$; fix a number $x$. Define $$X_{i}=\textbf{1}_{\{T_{i}\leq x\}}-F(x),\phantom{x} i=1,2,\ldots, $$ and $$M_{0}=0,$$ $$M_{n}=\sum_{i=1}^{n}X_{i}, \phantom{x} n\geq 1$$ a) Show…
Lech121
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Non-negative supermartingale: What is wrong with this proof?

Let $(X_t)$ be a non-negative supermartingale. Then for $s
Protawn
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Show there is a q such that $q^{S_n}$ is a martingale.

Consider a random walk on $\mathbb{Z}$ with increments $X_i$ that are some bounded random variable. Suppose $\mathbb{P}(X>0)$ and $\mathbb{P}(X<0)$ are non-zero, and $\mathbb{E} X\neq 0$. Show that there is some $0
Rann
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Second Equality of Wald

I'm doing an exercice about the second equality of Wald. Let $(X_i)_{i\ge 1}$ be a sequence of integrable random variables. Let $\mathcal{F} = (\mathcal{F}_i)_{i\ge 1}$ be a filtration such that $X$ is adapted. We suppose that $X_i$ and $F_{i-1}$…
Merli
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Can this attempt to prove Ito Isometry for Elementary Processes be fixed?

So I have found this link which I will try after writing this post, but I would like to see if my original attempt (which is his/her attempt there) can be made to work. The reason I want this to work is that this works for $M = $ Brownian…
Calvin Khor
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Under what conditions, if any, is a continuous function of a Martingale still a martingale?

Consider a martingale $M_s$ and a function $f(M_s) = \frac{1-e^{-kM_s}}{M_s}$. If $M_s$ is an Ito process, it is obvious from Ito's Lemma that $f(M_s)$ is not a Martingale. Does the restriction hold more generally (for non-Ito processes), or can…
Giorgi Chavchanidze
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Almost sure convergence of martingale increment

Prove or disprove. Suppose that $\left(M_{n}\right)_{n}$ is a martingale with $M_{n} \geqslant-10 \quad \forall n$, a.s. Is it true that $$ \sum_{i=1}^{\infty}\left(M_{i}-M_{i-1}\right)^{4}<\infty \quad \text { a.s. ? } $$ This was my attempt which…
Stochastichelp
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Prove $X_t=\sqrt{t} \sin B_t$ is a martingale, if $B_t$ is a Brownian motion

I previously asked whether $X_t=\sin B_t$ is a martingale, where $B_t$ is a standard Brownian motion and the underlined filtration is the canonical one. As it can be seen in the answer provided, this turned out to be false. The answer consisted of…
Barreto
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Is log(1+$M^{2}$) a maringale/submaringale/supermartingale, given that $M$ is a martingale?

Here it is what I have progressed: $Z_{t}\overset{def}{=}log(1+M^{2}_{t})$, Then there follows: $E(Z_t|\mathcal{F}_{s})=E\Big[log(1+M^{2}_{t})|\mathcal{F}_{s}\Big]\leq…
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