Questions tagged [martingales]

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

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18 views

Does the Martingale Representation theorem hold both ways?

Can the Martingale Representation theorem be used to assume that the integral with respect to Brownian motion, $B(t,\omega)$, $$X=\int^{T}_{0}B^{4}(t,\omega)dB(t,\omega)$$ is a square integrable ...
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16 views

Counterexample for continuous time sub-martingale convergence

We know that if $X_t$ is a right continuous sub-martingale with $\sup_t \mathbb{E}[X_t^+] < \infty$ then $\lim_{t \rightarrow \infty} X_t$ exists almost surely, but I haven't been able to find a ...
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22 views

Show that a L2 martingale with stationary, independent increments has a certain quadratic variation

I need help solving the following exercise out of chapter 1 of Karatzas, Shreve: 5.20 Exercise. Suppose $X\in\mathscr{M}_2$ has stationary, independent increments, and $(\mathscr{F}_t)_t$ is the ...
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Quadratic variation of true martingale

We know that for a continuous local martingale $M$ the quadratic variation $\left<M \right>$ is such that $M^2- \left< M \right>$ is a continuous local martingale. Is it true that if $M$ ...
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$L^p$ Boundedness of a martingale

So I recently read a paper where the authors claim that if for some martingale $(M_t)_{t\geq 0}$ we have $$\mathbb E[M_{t+s}^p]-\mathbb E[M_s^p]\leq \exp(-cs) (\mathbb E[M_{t}^p ]-\mathbb E[M_t]^p)$$ ...
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22 views

Describe all martingales that only take values ​in $\{−1, 0, 1\}$.

Describe all martingales that only take values ​​in $\{−1, 0, 1\}=:\Omega$. In the first instance i would try to find a filtration of $$P(\Omega)=\{\emptyset,\{0\},\{1\},\{-1\},\{1,-1\},\{1,0\},\{-1,...
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40 views

Stochastic processes - Why do we need filtration?

In the theory of stochastic process, besides the $\sigma$-algebra $\mathcal {F}$, we have an increasing sequence of $\sigma$-algebras $\{{\mathcal {F}}_{{t}}\}_{{t\geq 0}} $ called filtration. ...
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Non-uniqueness in the $L^1$ martingale representation

Let $\xi \in L^1(P,\mathfrak F_T)$ on some probability space with measure $P$, supporting a Brownian motion, we consider the augmented filtration $\mathfrak F$ associated to $W$, and a time $T>0$. ...
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40 views

Sum of independent random variables is a martingale which converges almost surely

Let $ \{X_n\}_{n \ge 1} $ be a sequence of independent random variables satisfying $$ \mathbb{P}(X_n = -n^2) = 1 - \mathbb{P}(X_n = \frac{n^2}{n^2 - 1}) = \frac{1}{n^2}. $$ The question is ...
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question about martingales [closed]

Let $\xi_{i}$ - i.i.d., $\mathbb{P}(\xi_{i} = 1) = \mathbb{P}(\xi_{i} = -2) = \frac{1}{2}$; $X_{i} = \sum_{j=1}^{i} \xi_{j}$. How to find the $\mathbb{P}(\exists n \geq 0: X_{i} = 1)$? Give me some ...
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About continuous local martingales, question on Le-Gall's book

Background Hello, I'm working on question 4.24 on Le-Gall's Brownian motion(...) and I would ask you to check if my ideas are correct. The question is as follows: $(M_t)$ is a cont. local ...
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15 views

Applyin Itô's formula in function of quadratic variation

I am learning some basic stochastic calculus and came across the following exercise: Consider a local martingale $M$ with continuous trajectories. Let $Z_t = \exp(M_t −0.5[M]_t)$. Show that Z ...
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Martingale problem about population of bacteria [closed]

I found a problem in Theory of Probability and Random Processes by Koralov and Sinai. Let $N_n,\ n \geq 1,$ be the size of a population of bacteria at time step $n$. At each time step each bacteria ...
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Martingales and OST examples (ABRACADABRA like)

The ABRACADABRA Problem is well known, now I need to understand examples that are like that approach. I think I know how to "calculate" the similar examples, but I would like to understand which ...
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1answer
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Verifying stochastic process containing summation of Martingales is a Martingale

Let $M=(M_t)_{t \geq 0}$ be a Martingale with respect to the filtration $\mathcal{F}=(\mathcal{F}_t)_{t \geq 0}$. Assume that $\mathbb{E}(M_t^2)<\infty$ for all $t \geq 0$. Let $0=t_0<t_1<...&...
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1answer
28 views

Unbounded quadratic variation process for bounded continuous martingale

Consider a bounded, continuous martingale $(X_t)_{t\ge 0}$. I was able to show that $(X^2-[X])_{t\ge 0}$ is uniformly integrable, where $[X]$ denotes the quadratic variation. Is there an example of a ...
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Is this solution of martingale problem correct?

I'm reading this problem and its solution I think there is a typo in the highlighted part. It should be $$\begin{array}{l} =\mathbb{E}\left[V_{n}^{2}\right]+\mathbb{E}\left[H_{n}^{2} \mathbb{E}\...
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25 views

Is it a good example for local martingale, but not for martingale?

If $B$ is a Brownian-motion in the $\mathcal{F}$ filtration, then the following process is a good example for being a local martingale, but not a martingale?$$S_{t}=\int_{0}^{t}\frac{1}{1-s}dB_{s},\;\;...
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1answer
27 views

Square integrable martingale bounded in $L^2$ if and only if the sum difference is square integrable

Let $\{Y_n\}_{n \ge 0}$ be a martingale with $ \mathbb{E}[Y_n^2] < \infty $ for all $n$. Show that $$ \sup_{n \ge 0} \mathbb{E}[Y_n^2] < \infty \Leftrightarrow \sum_{n \ge 0} \mathbb{E} [...
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42 views

The conditional expectation of iid random variables

Let $Y_1,Y_2,\dots,Y_n$ be a sequence of iid random variables. Each of them is integrable. Let $X_1=(Y_1+Y_2+\cdots+Y_n)/n,X_2=(Y_1+Y_2+\cdots+Y_{n-1})/(n-1),\dots,X_{n-1}=(Y_1+Y_2)/2,X_n=Y_1$ ...
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Showing some properties of local martingales in Karatzas, Shreve Problem 1.5.19

The goal is to show the following statements (given in Karatzas, Shreve, Chapter 1, 5.19 Problem): (i) A local martingale $X$ of class DL is a martingale. (ii) A nonnegative local martingale $...
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Expected time for hitting a distance in 2D random walk

Suppose $S_n$ is a 2D simple random walk starting at the origin moving up, down, left, and right with probabilities $\frac{1}{4}$ and $S_n$ is the vector in $\mathbb{Z}^2$ that denotes the position at ...
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1answer
36 views

Limit of the exponent of Random Variables

Suppose $X_1, X_2,\ldots$ are i.i.d. normal ($\mu=0, \sigma^2=1$) random variables and let $S_n$ denote the sum of first $n$ $X_i$'s. Show that $$\lim_{n\to \infty} \exp(2S_n - 2n)=0$$ I think I am ...
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71 views

Pólya's Urn Long-Term Probability

I am tasked with the following problem: Let $M_n$ be the fraction of white balls in Pólya's urn after $n$ draws, where you start with one white and one black ball, and after every draw, you add ...
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32 views

Positive continuous supermartingale is a proper martingale

Let $M$ be continuous positive supermartingale with $\mathbb{E}[M_0]< \infty$. By the supermartingale convergence theorem $M_\infty = \lim M_t$ exists almost surely. How do I show that, if $\mathbb{...
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1answer
31 views

Properties of Martingales [closed]

I found following problems about properties of martingales and I would like to know is my approach correct for first problem and how to exactly solve second one. Problem is following: Let $\{Y_n\}_{n\...
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1answer
41 views

Given a sequence of i.i.d. random variables, prove a result involving conditional expectation by means of symmetry argument

Let $(X_n)_{n\geq1}$ be an i.i.d. sequence with $\mathbb{E}\{|X_1|\}<\infty$. Let $S_n=X_1+\cdots+X_n$ and $\mathcal{F}_{-n}=\sigma(S_n,S_{n+1},\ldots)$. Then, one can state that $$M_{-n}=\mathbb{E}...
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If $(X_n)_{n \ge 0}$ is a martingale and $E[\tau_n - \tau_{n-1}]=E[(X_n-X_{n-1})^2]$ then $E[\tau_n]=E[X_n^2]$?

Given a square-integrable martingale $(X_n)_{n \ge 0}$ and a sequence of stopping times such that $$ E[\tau_n - \tau_{n-1}]=E[(X_n-X_{n-1})^2] $$ I'm trying to verify the following statement: ...
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1answer
96 views

Some doubts in a part of the proof of Backwards Martingale Convergence Theorem (Jacod-Protter)

A USEFUL RESULT (Doob's Upcrossing Inequality) Let $(X_n)_{\geq0}$ be a submartingale, let $a<b$ and let $U_n$ be the number of upcrossings of $[a,b]$ before time $n$. Then $$ \mathbb{E}\{U_n\}\...
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16 views

Given a standard Brownian motion $W_t$, (i) what are the distributions of $|W_t|$ and $tW_t$? (ii) Are they martingales?

For (i), I believe I am correct in saying that $|W_t|$ is a folded normal distribution with mean $\sqrt{\frac{2}{\pi}}t$ and variance $t$, and $tW_t$ is distributed normally with $\mathbb{E}(tW_t)=0, \...
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35 views

Given a martingale $Y=\mathbb{E}\{M_n|\mathcal{F}_n\}$, why does the below red inequality hold true?

Given a probability space $(\Omega$, $\mathcal{F}$, $\mathbb{P})$, let $Y$ be a random variable defined on it such that $Y\in \mathcal{L}^1$ and $M_n=\mathbb{E}\{Y|\mathcal{F}_n\}$ be a martingale. ...
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114 views

Show martingality property for r.v. $S_{\infty}$, given some assumptions. Could you please detail your answer to my two points?

Given a probability space $(\Omega$, $\mathcal{F}$, $\mathbb{P})$, let $(S_n)_{n\geq1}$ be a martingale and a sequence of uniformly integrable random variables. Also assume $S_n\rightarrow S_{\infty}$ ...
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$X=M+A$ bounded semimartingale, then $M$ and $A$ are bounded? Counterexample?

Let be $X=M+A$ a semimartingale with $M$ being a local martingale and $A$ an adapted process a finite variation. If $M$ and $A$ are bounded, then of course $X$ is bounded as well. Is the converse true?...
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$\sigma$-algebra of the past intersection property

Let $(\Omega,\mathcal{F},\mathcal{F}_{\cdot}=(\mathcal{F}_k)_{k\in\mathbb{N_0}},\mathbb{P})$ be a filtered probability space and $\sigma,\tau:\Omega\to\mathbb{N}\cup\{\infty\}$ two $\mathcal{F}_{\cdot}...
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34 views

How to prove $\mathbb{E}\{M_m 1_{\Psi}\}=\mathbb{E}\{M_n1_{\Psi}\}$, given that $n\geq m$, $\Psi\in\mathcal{F_m}$ and $(M_n)_{n\geq1}$ is martingale?

Given a probability space $(\Omega$, $\mathcal{F}$, $\mathbb{P})$, $(M_n)_{n\geq1}$ is a martingale. Let $\Psi \in \mathcal{F}_m$ and $n\geq m$. How can one prove that, by martingale property \begin{...
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1answer
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A question about proof of Martingale Convergence Theorem. Why does the Uniform integrability imply the following fact?

Relying on the below definition of uniform integrability: Definition: A subset $\mathcal{U}$ of $\mathcal{L}^{1}$ is said to be a uniformly integrable collection of random variables if \begin{...
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$Var (X_n)= Var(X_0) +\sum_{k=0}^{n} Var (X_{k}-X_{k-1})$, Martingale Variance

i'm already diving through the martingales properties and try to prove the following problem: Let $X_{n}, Y_{n}$ two martingales squared integrable respect to the filtred probability space $(\Omega, ...
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1answer
31 views

Optional sampling theorem St. Petersburg paradox

I want to show that the optional sampling theorem does not hold for unbounded stopping times using the example of the St. Petersburg paradox/St. Petersburg game. We have a consecutive (fair) coin toss ...
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1answer
22 views

Show this statement holds for a right continuous martingale.

Let $T > 0$ and consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ on which there is a filtration $$\{\mathcal{F}_t: 0 \leq t \leq T\}$$ Let $M:=\{M_t: 0 \leq t \le T\}$ be a ...
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1answer
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A doubt on the proof of Martingale Convergence Theorem on Jacod-Protter

Theorem: Let $(X_n)_{n\geq1}$ be a submartingale such that $\sup\limits_{n}\mathbb{E}\{X_n^{+}\}<\infty$. Then, $\lim\limits_{n\rightarrow\infty} X_n = X$ exists a.s. (and is finite a.s.). Moreover,...
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$\mathbb{E}((M(\tau)-M(\rho))^2|F_{\rho})=\mathbb{E}(M^2(\tau)-M^2(\rho)|F_{\rho})$

$\rho,\tau$ - stopping times such that $0\le \rho\le\tau\le T$ and $M(t),F_t)$ square integrable martingale. Prove that $\mathbb{E}((M(\tau)-M(\rho))^2|F_{\rho})=\mathbb{E}(M^2(\tau)-M^2(\rho)|F_{\rho}...
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1answer
19 views

Why the process $n \mapsto e^{-aT_n}f(X_{T_n})$ is a super martingale.

Let $M := (e^{-aT_n}f(X_{T_n}); \mathcal{F}_n)$, where $a$ is constant, $\mathcal{F}_n$ is appropriately defined filtration and $T_n$ is $n$th jump of an independent (independent of $X_t$) Poisson ...
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39 views

$\text{lim inf}_{t \downarrow 0}\frac{\int_{0}^{t}E[X_{s}^{2}]ds}{t^{2}}\leq \text{lim sup}_{t \downarrow 0}\frac{\int_{0}^{t}E[X_{s}^{2}]ds}{t^{2}}$

Let $g(t) = E[X_{t}^{2}]$, the starting point is the following set of inequalities which hold for all $t \geq 0$: \begin{align} - \int _{0}^{t} 2g(s)ds + t \leq g(t) \leq \int _{0}^{t} 2g(s)ds + t \...
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13 views

I am trying to construct a process $Y$ in the martingale representation of the following two terms

(i) $M_t = \mathbb{E}[B_t^2|F_t]$ (ii) $M_t = \mathbb{E}[\text{max}\{B_t,0\}|F_t]$ Where $B$ is a one-dim. Brownian Motion and $F$ its $P$-completed canonical filtration. I think in (ii) we can ...
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1answer
26 views

Prove that process is a local martingale.

$ \{X_{t}\}_{t\geq 0}, \{Y_{t}\}_{t\geq 0} $ - Ito processes. From Ito's formula we have got: $$ X_{t}Y_{t} = X_{0}Y_{0} + \int_{0}^{t}X_{s}dY_{s} + \int_{0}^{t}Y_{s}dX_{s} + \int_{0}^{t}dX_{s}dY_{s} ...
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1answer
24 views

Why is the fact that the sequence $(M_n)_{n\geq0}$ is increasing shown in the following way? My alternative

I quote Jacod-Protter Given a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ and an increasing sequence of $\sigma$-algebras $\left(\mathcal{F}_n\right)_{n\geq0}$, let $M=\left(M_n\...
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1answer
23 views

Optional sampling theorem for bounded stopping times in continuous time

Establish the optional sampling theorem for a right-continuous submartingale $\{X_{t},\mathcal{F}_{t}, t \in [0,\infty)\}$ under either of the following conditions: $a)$ $T$ is a bounded optional ...
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92 views

Asymmetric Random Walk with Absorbing Barriers as a Martingale

Let $Xn$ be a random walk with absorbing barriers on state space $S = \{0, 1, \ldots, 15\}$ with probability $p=\frac{3}{4}$ of moving to the right and $1-p=\frac{1}{4}$ for moving to the left. Let $f:...
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35 views

Doubt on definition of Doob's notion of upcrossing

From Jacod-Protter: DEFINITION Let $(X_n)_{n\geq 0}$ be a submartingale and $a<b$. The number of upcrossings of an interval $[a,b]$ is the number of times a process crosses from below $a$ and to ...
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17 views

Multidimensional Ito Formula - martingale

Hey I have $W_t=(W_t^1,...,W_t^n)$ - n-dimensional Wiener process, $f\in C^2$, $f:\mathbb{R}^n\to \mathbb{R}$, and for every $x=(x_1,...x_n)$ I have $f(x+W_t)=f(x)+\sum_{i=1}^{n}\int_0^t \frac{\...

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