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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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martingale expected value

By definition of a martingale $\{Y_n\}$ with respect to to a sequence $\{X_n\}$, $\text{E}(Y_{n+1}|X_1,\ldots,X_n) = Y_n$. According to my teacher, it is also true that $\text{E}(Y_{n+a}|X_1,\ldots,X_n) = Y_n$, for any integer $a \geq 1$. (1) This…
user63010
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Show that $f(W_{t})$ is submartingale <=> $f$ is convex.

Show that $f(W_{t})$ is submartingale ($W_t -$ Brownian motion) $\iff f$ is convex. I can only show one way: $\Leftarrow$ $f(W_{s})=f(\mathbf{E}(W_t|F_s))\leq \mathbf{E}(f(W_t)|F_s)$ and from definition $f(W_{t})$ is a submartingale.
asparagus27
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Martigale of Gambler's Ruin

Let $P\left ( X_{i}=1 \right )=p$ and $P(X_{i}=-1)=1-p$ where $p \in \left ( 0,1 \right )$ and $p\neq \frac{1}{2}$. Let $S_{n}=S_{0}+X_{1}+\cdot \cdot \cdot X_{n}$. Define $M_{n}=\left ( \frac{1-p}{p} \right )^{S_{n}}$ Recall: if $h\left ( x \right…
Mathematicing
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What does it mean to say a martingale is closed?

What does it it mean to say a martingale is closed [in layman terms] ? Does it just mean the martingale converges ??
user1769197
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Why are martingales always adapted to a filtration?

Let $(M_t)$ a martingale and $(\mathcal F_t)$ a filtration. Then, $\mathbb E|M_t|<\infty $ for all $t$ and $$\mathbb E[M_t\mid \mathcal F_s]=M_s,$$ when $s\leq t$. Why this last condition tell us that it's always adapted to $\mathcal F_t$ ? (i.e.…
user380364
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Martingales and Stochastic Analysis James Yeh Th 8.13

Can someone check if the proof of theorem 8.13 of the book Martingales and Stochastic Analysis by James Yeh is correct (link here: https://goo.gl/ivxJnv, the Google Book version). Note line 11, page 135 - "On the set $\{S \leq t \}$, we have…
Project Book
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Why is right-continuity important in the martingale convergence theorem

Let $(X_t)$ be a right-continuous super-martingale such that $\sup_t E[X_t^-] < \infty$. Then $\lim_{t \to \infty} X_t = X_\infty$ a.s. where $X_\infty$ is integrable. I am trying to prove this. I know the proof for the countable case, i.e. if…
jpv
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Prove this is a Martingale

Prove that $$Z_t:=\frac{e^{W_t^2/(1+2t)}}{\sqrt{1+2t}}$$ is a $\mathscr{F}_t$-martingale. I have tried all the usual manipulations without any success. The only useful fact I think should be used is that: if $X_t\sim N(\mu,\sigma^2)$…
mastro
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Martingale convergence

I'm working on the same problem on this page except for the fact that my conditional expectation is $E[X_{n+1}|F_{n}]\leq X_{n}+Y_{n}$ I can't find a RV Z to recover a supermartingale from the equation.. I need some help.. Let $X_n$ and $Y_n$ be…
sky90
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P and Q martingales

Let $\mathbb{P},\mathbb{Q}$ be two equivalent probability measures. Could someone come up with an example of a $\mathbb{P}$-martingale, which is not a $\mathbb{Q}$-martingale and another example of a martingale which is both a $\mathbb{P}$ and…
Tob4U
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Minimum submartingale inequality

I have a problem that goes like this: Let $(X_n)_{n\geq 0}$ be a submartingale and the constant $\lambda>0$. Show that $\lambda\mathbb{P}(\min_{0\leq k\leq n}X_k<-\lambda)\leq\mathbb{E}[X_n]-\mathbb{E}[X_0]$. My attempt is the following: Define a…
Tob4U
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$P(i)$ the probability that Markov chain eventually enters state $N$ given that it starts in state $i$. Show $\{P(X_n), n\geq 0\}$ is a martingale

Consider the Markov chain $\{X_n, n \geq 0\}$ with $P_{NN} = 1$. Let $P(i)$ denote the probability that this chain eventually enters state $N$ given that it starts in state $i$. Show that $\{P(X_n), n\geq 0\}$ is a martingale. This is exercise 6.5…
Math_Day
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Marcinkiewicz-Zygmund type inequality

if we consider an independent sequence of random variables $X_1,X_2,...$ and denote with $S_n = \sum_{j=1}^n X_j$ the sequence of partial sums, one can easily verify the Marcinkiewicz-Zygmund inequality: $\Vert S_n \Vert_2^2 \leq \sum_{j=1}^n \Vert…
student7481
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Filtration and martingale from definition

On $([0,1], \mathcal{B}(0,1), \lambda)$ (Lebesgue measure) consider random variables $Y_n(\omega) = \omega^21_{[0,1-1/n]} + 1_{(1-1/n,1]}$ and $X(\omega) = 2\omega$. I have following questions to answer Find the filtration generated by $Y_n$. My…
Barabara
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Martingale example

I am new on martingales and I try to understand the following example: Why we use there that $S_{n-1}$ is $\mathcal{F}_{n-1}$ measurable and not directly that $S_n$ is $\mathcal{F}_n$ measurable? Furthermore how can we conclude that $\mathbb…
Frederick Manfred
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