All Questions
Tagged with pr.probability martingales
210 questions
2
votes
0
answers
121
views
An unnatural martingale
What is an example of a real valued stochastic process $X$, and a filtration $\mathcal F_t$ such that $X$ is a martingale with respect to $\mathcal F_t$ but not it’s natural filtration?
Either ...
- 6,195
0
votes
1
answer
315
views
When is every Levy martingale of a process a continuous martingale?
Let $X_t$ be a real valued stochastic process, and $\mathcal H_t$ the the natural filtration of $X_t$.
Under what conditions on $X$ does the following statement hold?
For every $\mathcal H_\infty$-...
- 6,195
12
votes
0
answers
196
views
UMD constant of finite dimensional spaces
For a Banach space $B$, its one-sided Unconditional Martingale Difference (UMD) constant $C^-_p$ (for $p \in (1,\infty)$) is the smallest value such that for all $B$-valued martingale difference ...
- 408
1
vote
0
answers
47
views
$\exists c \in\mathbb{R}_+^*,\forall p,r\in \mathbb{R}_+,E[|X_{p+r}-X_r||\mathcal{F}_r] \leq c$ implies the optional stopping theorem
Consider a integrable submartingale $(X_r)_{r \in \mathbb{R}_+}$ relative to $(\mathcal{F}_{r})_{r \in \mathbb{R}_+}$ and such that $$\exists c \in \mathbb{R}_+^*,\forall k \in \mathbb{N},E[|X_{k+1}-...
- 249
2
votes
1
answer
638
views
$L^p$-convergence of submartingale
Let $p\geq1.$ Consider a $\mathcal{F}_k$-submartingale $(X_k)_k$ in $L^p.$ We can prove easily that $(X_k)_k$ converges in $L^p$ if and only if $(|X_k|^p)_k$ is uniformly integrable.
If $(X_k)_k$ was ...
- 249
6
votes
3
answers
999
views
Does there exist an almost surely differentiable martingale?
Does there exist a continuous time martingale $X_t$ not a.s. constant in $t$ that is almost surely everywhere differentiable?
- 6,195
1
vote
0
answers
53
views
A semimartingale interpolation problem
This question is a direct extension of this one.
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a stochastic basis and let $N\in\mathbb{Z}^+$, $T>0$, $\{t_n\}_{n=1}^{N}$ be a ...
- 77
1
vote
1
answer
140
views
Does a sequence that verifies the assumptions of a square integrable martingale on some event need to be convergent on this event?
I came across this claim by reading some literature on stochastic approximation.
Let $(\Omega, \mathcal{A}, \mathbb{P}$) be a probability space, $(\mathcal{F}_n)$ a filtration on it. Let $(\epsilon_{n}...
- 115
2
votes
1
answer
148
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If a process is periodic on average with mutually incommensurable periods, is the process a martingale?
Motivation:
If a continuous function on the real line is periodic with periods $p_1, p_2 > 0$ such that $\frac{p_1}{p_2}$ is irrational, then the function is constant. Is there a probabilistic ...
- 6,195
4
votes
1
answer
677
views
If the moving average of a process is a martingale, is the process a martingale?
Problem set up:
Let $\mathcal F_t$ be a filtration satisfying the usual conditions. Let $T > 0$ be a fixed real number, and define the filtration $\mathcal H_t := \mathcal F_{T + t}$.
Suppose a ...
- 6,195
1
vote
1
answer
109
views
Weaker than martingale condition
Let $\mathcal{F}_n$ be a filtration and $S_n$ be a sequence such that $\mathbb{E}[S_n-S_{n-1}|\mathcal{F}_{n-2}]=0$ for all $n$. This condition is similar to the martingale condition but the ...
- 31
1
vote
0
answers
108
views
Decomposition of reversed processes
Consider a reversed filtration $(\mathcal{F}_k)_{k \geq 0} $ $(\mathcal{F}_{k+1} \subset\mathcal{F}_k),$ $(X_k)_{k \geq0}$ is a processes in $L^1,\mathcal{F}_k$-adapted.
Is it possible to decompose $...
- 249
1
vote
0
answers
80
views
Almost supermartingale and a.s convergence
After reading a paper on the convergence of almost supermartingale, the following result appeared:
If $(X_k)_k,(Y_k)_k,(W_k)_k$ are three $(\mathcal{F}_k)$-adapted processes taking values in $\mathbb{...
- 249
0
votes
0
answers
188
views
Moment generating function of a stopped process from Wald's identity
In an exercise I am asked to prove the following Wald's identities: let $S_n$ be a simple random walk and $T$ a stopping time. Then for all $\lambda \in \mathbb R,$
$$
\mathbb E(e^{\lambda S_1}) = 1 \...
- 1,755
3
votes
0
answers
81
views
How can we use Martingales to identify an unknown particle?
Suppose there is a particle in a box. We are interested in identifying what type of particle it is, but are not allowed look inside the box. All we can do is observe the particles that are entering ...
- 1,955
1
vote
0
answers
526
views
Martingales associated with heat equation
I am trying to learn the connection between Brownian motion and heat equation (in the spirit of Feynman-Kac, for example, here). I read (Michael E. Taylor's PDE book, Volume II, Chapter 11, ...
- 1,407
1
vote
1
answer
182
views
Is a stopped Ito-integral integrable if the Ito integrand is only square-integrable on an open interval?
Assume a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\in[0;T)}, \mathbb P)$ with an $\mathbb R^n$-valued Brownian motion $\{W_t\}_{t\in[0;T)}$ and the filtration $\{\mathcal F_t\}_{t\in[0;T)...
- 335
0
votes
0
answers
71
views
Conditions for existence of a semi-martingale representing a system of probability measures
Let $(\nu_t)_{t \in [0,1]}$ be Borel probability measures on a stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_{t \in [0,1]})_t,\mathbb{P})$.
Does there exist a semi-martingale $(X_t)_{t\in[0,1]}$ ...
- 5,405
1
vote
0
answers
744
views
Local martingale but not martingale
For a 3-dimensional Brownian motion $B = (B_t, t ≥ 0)$ and $x ∈ \mathbb{R}^3 \backslash \{0\}$ define the process
$Y = (Y_t, t ≥ 0)$ via $Y_t =\frac{1}{|B_t+x|}$ how come this is a continuous local ...
1
vote
0
answers
43
views
Understanding the space of parameters in a covariance matrix of conditional expectations
Let $\{(Y_n, Z_n)\}_{n=-\infty}^{n=\infty}$ be a zero-mean jointly stationary Gaussian process where $Z$ takes values in $\mathbb{R}$ and $Y$ takes values in $\mathbb{R}^k$. Here, $n$ runs over the ...
- 1,269
10
votes
4
answers
679
views
The min of the mean of iid exponential variables
Let $X_1, \ldots, X_n, \ldots$ be iid exponential random variables with mean 1. It is well-known that $\min_{1\le j < \infty} \frac{X_1 + \cdots + X_j}{j}$ follows the uniform distribution U(0,1). ...
- 773
2
votes
0
answers
237
views
Semimartingale decomposition and filtrations
In short: I am trying to understand how the decomposition of a semimartingale into its local martingale and finite variation components depends on the filtration we are using.
So, taking a toy example,...
- 341
3
votes
2
answers
635
views
Exponential inequality for the sum of martingale differences $X_1, \dots, X_n$ when $\sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2$
Let $X_1, X_2, \dots, X_n$ be a martingale difference sequence such that
$$
X_i \leq y \quad \text{and} \quad \sum_{i=1}^{n} \operatorname{Var}(X_i) \leq B^2.
$$
Question 1: Does the following hold?
$$...
- 33
1
vote
1
answer
284
views
Martingale derivation by direct calculation
I'm reading the proof of a theorem and stumbled across the following derivation which I cannot replicate myself.
Let $W(t)$ be a $Q$-martingale and be given by $W(t) = B(t) + \mu t$ with $B(t)$ a ...
- 11
13
votes
1
answer
713
views
Identity involving the probability that a random walk stays below a curve
I'm looking for a direct proof of the following identity:
Let $W_n$ be a simple random walk with $W_0=0$. For all $x>0$ we have
$$
\lim _{N\to \infty} \sqrt{N} \cdot \mathbb P \Big( \forall n \le ...
- 723
2
votes
1
answer
144
views
English translation of "Une inégalité pour martingales à indices multiples et ses applications"
Does anyone know of a English translation of "Une inégalité pour martingales à indices multiples
et ses applications" by Renzo Cairoli. Or could translate the statement of the martingale ...
- 243
2
votes
1
answer
300
views
On the speed of divergence of the converse of the Strong law of large numbers
By the converse of the strong law of large numbers, we know that, given a sequence of i.i.d random variables $X_1,X_2,\dots$ such that $\mathbb{P}(X_1 \ge 0)=1$ and $\mathbb{E}X_1= \infty$,
then I ...
- 446
6
votes
1
answer
421
views
Probability in Chromatic number upper bound of induced subgraph
Let $G=(V, E)$ be a graph with chromatic number $\chi(G)=1000 .$ Let $U \subset V$ be a random subset of $V$ chosen uniformly from among all $2^{|V|}$ subsets of $V$. Let $H=G[U]$ be the induced ...
- 61
0
votes
1
answer
2k
views
Martingale convergence theorem in Polya's urn
I want to get checked if my attempt is okay.
First off, let me shortly describe what Polya's urn is:
A certain urn initially contains a red and a blue ball. We now repeatedly do the following : we ...
7
votes
2
answers
2k
views
Proof of extended supermartingale convergence theorem
There is a supermartingale convergence theorem which is often cited in texts which use Stochastic Approximation Theory and Reinforcement Learning, in particular the famous book "Neuro-dynamic ...
- 183
5
votes
1
answer
165
views
Is there an i.i.d sequence in the unit cube $[-1,1]^d$ with $\mathbb E \left[ \Big \| \sum_{i=1}^N X_N \Big \|_\infty\right] = \sqrt {dN}$?
There are loads of concentration results for sums of scalar-valued independent sums $X_1,X_2,\ldots, X_N$ with $\mathbb E[X_n]=0$. For example Hoeffding's Inequality says if all $|X_1|\le 1$ then $\...
- 1,955
0
votes
2
answers
251
views
Martingale optional stopping before a stopping time
Here’s an easy one, I hope:
Suppose $\tau$ is a stopping time and $(M_t)$ is a martingale which together satisfy the hypotheses of the optional stopping theorem so that $\mathbb{E}[M_\tau]= \mathbb{E}...
- 3
6
votes
0
answers
150
views
Delayed Pólya's urn process
The standard Pólya's urn process can be stated as follows:
You have an urn with red and green balls. At any time unit you choose one ball at random, note the colour, and give the ball back. At the ...
- 396
1
vote
1
answer
361
views
Length of longest subsequence as a martingale
Consider a sequence of continuous random variables $(X_n)_{n \geq 1}$. Let $Y_n$ denote the longest increasing subsequence in the tuple $(X_1,\dots,X_n)$. Does $Y_n$ form a martingale? If not, can I ...
- 1,412
1
vote
0
answers
393
views
Expected number of games for three-player gambler's ruin
Three gamblers each start with $a$, $b$ and $c$ chips, respectively.
In each round of the game, a gambler is selected uniformly at random
to give up one chip, and one of the remaining two gamblers is ...
- 1,412
5
votes
1
answer
208
views
Expected supremum of normalised random walk
Let $X^i\in \mathbb R^d$ be iid. random variables for $i=1$ to $n$.
Assume $\mathbb E[X^i]=0$ and the covariance matrix $\mathbb C[X^i] = \mathbb E[X^iX^{iT}] = I$ is the identity matrix.
Define $S^k=...
3
votes
0
answers
132
views
Embedding a continuous-time martingale in Brownian motion
Using the Skorohod embedding, we can embed any square-integrable discrete time martingale $(M_n)$ into a Brownian motion, obtaining times $(T_n)$ such that $(B(T_n))_{n\ge 0}$ is a version of $(M_n)$. ...
- 349
1
vote
0
answers
265
views
Wiener isometry for semimartingales
Suppose that $Y_t$ is a special square-integrable $\mathbb{R}$-valued semi-martingale and let $\mathcal{L}^2(Y)$ denote the set of $Y$-predictable processes satisfying
$$
\mathbb{E}\left[
\int_0^{\...
- 5,405
4
votes
1
answer
594
views
Martingales and intersection of random walks
Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...
- 519
2
votes
1
answer
287
views
Bernstein Inequality for continous local martingale
I'm looking for a simple proof of the following fact, which is somehow Bernstein inequality in continuous time.
Let $(M_t)_{t\geq 0}$ be a continuous local martingale. Then :
$$P\left(\sup_{t\in [0,...
- 245
4
votes
0
answers
143
views
For a martingale $f_0,f_1,\ldots $ how can we bound $P(\frac{1}{n} \|f_n\| \le 1$ for all $ n \ge N)$?
Suppose $f_0,f_1, \ldots$ is a martingale (or i.i.d sequence) in $\mathbb R^d$ with $f_0=0$ and all $\|f_n - f_{n-1}\| \le L$ say. There are many concentration results for the initial segment of the ...
- 1,955
7
votes
1
answer
409
views
Do i.i.d. sums concentrate any faster than martingales?
Suppose $X_1,X_2, \ldots, X_N \in \mathbb R^d$ are random variables with each $\|X_n\|_2 \le 1/2$ (this choice of the constant simplifies later formulae).
The simplest concentration inequality I know ...
- 1,955
2
votes
1
answer
161
views
Concavity, martingales and stopping time
Suppose $(x_t)_t$ is a bounded $\mathbb F_t$ martingale and $f(t,x)$ is continuous, bounded, and concave in $x$. So, for any $s \ge t$, $$\mathbb E_t f(s,x_s) \le f(s,\mathbb E(x_s)) = f(s,x).$$
Does ...
- 553
0
votes
1
answer
111
views
Conditioning on an irrelevant variable in a martingale control problem
Suppose I have two independent Brownian motions $B^1_t, B^2_t$ and $\mathbb F_t$ be the natural filtration generated by them. Let $T > 0$ be a fixed finite number. Let $q_t$ be a $[-1,1]$ valued $\...
- 553
12
votes
2
answers
2k
views
Can we do better than Azuma-Hoeffding when the variance is small?
The Azuma-Hoeffding Inequality says that if $X_1,X_2, \ldots$ is a martingale and the differences are bounded by constants, $\|X_i - X_{i-1}\| \le 1$ say, then we should not expect the difference $\|...
- 1,955
1
vote
0
answers
58
views
Martingales limit theorems (reference)
I have a sequence of processes $\{X^N(t)\}_{t\in [0,T]}$, $N\in\mathbb N$ such that
$X^N(t)=x+M^N(t)$,
where $M^N(t)$ is a martingale with expectation $0$ and with quadratic variation $<M^N>(t)$ ...
- 139
3
votes
0
answers
75
views
p-Variation distance defines semi-martingales
Question
When, does the process $\tilde{X}_t$, defined path-wise by
$$
\tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right),
$$
define a ...
- 5,405
3
votes
2
answers
229
views
Expectation of the exitpoint distance for the symmetric random walk
Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$.
Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\dotsb+X_n$, ...
- 31
3
votes
1
answer
237
views
Concentration of a modified random walk
Let $\varepsilon$ be a number in $(0, 1)$, consider the following random walk on the real line $X^{(0)}, X^{(1)}, \dots$, where
$X^{(0)}=0$
If $X^{(t)} > 0$, then with probability $.5$, $X^{(t+1)...
- 143
3
votes
1
answer
177
views
Convergence of SDEs
Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...
- 5,405