Questions tagged [martingales]
For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.
2,433
questions
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1answer
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 ...
2
votes
0answers
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 ...
1
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0answers
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 ...
3
votes
0answers
26 views
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 $M$ ...
0
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1answer
33 views
$L^p$ 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
$$\mathbb E[M_{t+s}^p]-\mathbb E[M_s^p]\leq \exp(-cs) (\mathbb E[M_{t}^p ]-\mathbb E[M_t]^p)$$
...
0
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1answer
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
$$P(\Omega)=\{\emptyset,\{0\},\{1\},\{-1\},\{1,-1\},\{1,0\},\{-1,...
2
votes
1answer
40 views
Stochastic processes - Why do we need filtration?
In the theory of stochastic process, besides the $\sigma$-algebra $\mathcal {F}$, we have an increasing sequence of $\sigma$-algebras $\{{\mathcal {F}}_{{t}}\}_{{t\geq 0}} $ called filtration.
...
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10 views
Non-uniqueness in the $L^1$ martingale representation
Let $\xi \in L^1(P,\mathfrak F_T)$ on some probability space with measure $P$, supporting a Brownian motion, we consider the augmented filtration $\mathfrak F$ associated to $W$, and a time $T>0$. ...
1
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0answers
40 views
Sum of independent random variables is a martingale which converges almost surely
Let $ \{X_n\}_{n \ge 1} $ be a sequence of independent random variables satisfying
$$
\mathbb{P}(X_n = -n^2) = 1 - \mathbb{P}(X_n = \frac{n^2}{n^2 - 1}) = \frac{1}{n^2}.
$$
The question is ...
-1
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0answers
27 views
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 ...
3
votes
1answer
32 views
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 ...
0
votes
1answer
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 ...
-1
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0answers
20 views
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 ...
0
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0answers
21 views
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 ...
2
votes
1answer
34 views
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<...&...
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 ...
0
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0answers
21 views
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}\...
1
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0answers
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?$$S_{t}=\int_{0}^{t}\frac{1}{1-s}dB_{s},\;\;...
0
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1answer
27 views
Square integrable martingale bounded in $L^2$ if and only if the sum difference is square integrable
Let $\{Y_n\}_{n \ge 0}$ be a martingale with $ \mathbb{E}[Y_n^2] < \infty $ for all $n$. Show that
$$
\sup_{n \ge 0} \mathbb{E}[Y_n^2] < \infty \Leftrightarrow \sum_{n \ge 0} \mathbb{E} [...
0
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0answers
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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0answers
23 views
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 $...
2
votes
0answers
37 views
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 ...
0
votes
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 ...
2
votes
2answers
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 ...
4
votes
2answers
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{...
0
votes
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\...
1
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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 $$M_{-n}=\mathbb{E}...
1
vote
2answers
40 views
If $(X_n)_{n \ge 0}$ is a martingale and $E[\tau_n - \tau_{n-1}]=E[(X_n-X_{n-1})^2]$ then $E[\tau_n]=E[X_n^2]$?
Given a square-integrable martingale $(X_n)_{n \ge 0}$ and a sequence of stopping times such that
$$
E[\tau_n - \tau_{n-1}]=E[(X_n-X_{n-1})^2]
$$
I'm trying to verify the following statement:
...
1
vote
1answer
96 views
Some doubts in a part of the proof of Backwards Martingale Convergence Theorem (Jacod-Protter)
A USEFUL RESULT (Doob's Upcrossing Inequality)
Let $(X_n)_{\geq0}$ be a submartingale, let $a<b$ and let $U_n$ be the number of upcrossings of $[a,b]$ before time $n$. Then
$$
\mathbb{E}\{U_n\}\...
1
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0answers
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, \...
0
votes
2answers
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. ...
2
votes
2answers
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 $S_n\rightarrow S_{\infty}$ ...
1
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0answers
14 views
$X=M+A$ 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?...
0
votes
2answers
22 views
$\sigma$-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}...
1
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0answers
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{...
1
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1answer
85 views
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{...
1
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0answers
21 views
$Var (X_n)= Var(X_0) +\sum_{k=0}^{n} Var (X_{k}-X_{k-1})$, 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, ...
1
vote
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 ...
0
votes
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 ...
0
votes
1answer
15 views
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,...
1
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2answers
25 views
$\mathbb{E}((M(\tau)-M(\rho))^2|F_{\rho})=\mathbb{E}(M^2(\tau)-M^2(\rho)|F_{\rho})$
$\rho,\tau$ - 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}...
0
votes
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 ...
1
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0answers
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
\...
0
votes
0answers
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 ...
0
votes
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} ...
0
votes
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\...
0
votes
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 ...
1
vote
0answers
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:...
0
votes
0answers
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 ...
0
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0answers
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{\...
