All Questions
Tagged with pr.probability martingales
210 questions
3
votes
1
answer
140
views
Lindeberg implies convergence of max of conditional variances in L1
The following is taken from Dvoretzky, 1972, ASYMPTOTIC NORMALITY FOR SUMS OF DEPENDENT RANDOM VARIABLES, Equation 4.6.
$$\{X_{n,k}\}_{n=0,1,...;k=0,1...,k_n}$$
is a (triangular) array of r.v.'s /w
...
- 115
3
votes
2
answers
517
views
CLT for Martingales
I posted this question originally in math stack exchange, but I got no answer.
(https://math.stackexchange.com/questions/2604591/clt-for-martingales)
In wikipedia, there is a version of a CLT for ...
- 339
3
votes
1
answer
267
views
An identity for the exponential of a martingale
I am trying to understand a Lemma in Olav Kallenberg's book "Foundations of Modern Probability" (Lemma 26.19 in the second edition or 23.19 in the first edition).
The part of the lemma that I do not ...
3
votes
1
answer
554
views
A concentration inequality derived from Freedman’s inequality
Freedman’s inequality is a well-known concentration inequality of martingale difference sequence:
Let $(Z_t)_{t \leq T}$ be a real-valued martingale difference sequence adapted to filtration $\...
- 61
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
3
votes
0
answers
101
views
Divergent/Unbounded random walks techniques
I want to prove the following biased random walk will be diverge. Suppose I have a random walk $S_n = X_1 + ... + X_n$, but $X_1,...,X_n$ are dependent variables. $X_1 \sim$ Bernoulli($\sigma(\theta_1)...
- 31
3
votes
0
answers
80
views
Seeking strong bounds on KL-divergence and martingales for a hypothesis-testing inequality
Let's say we have a finite set $\mathcal{O}$ of observations, and let $\mathcal{C}(\Delta\mathcal{O})$ denote the space of closed convex sets of probability distributions.
We have two hypotheses which ...
- 353
3
votes
1
answer
181
views
When does a local supermartingale become a proper supermartingale?
This is a cross-post of my question on MSE.
Abstract: When a local supermartingale is bounded from below, is it a proper supermartingale?
Question: In remark 4.2 (p.16) of the lecture notes by Martin ...
- 235
3
votes
0
answers
147
views
Request for article in Rev. Roumaine Math. Pures Appl. (1981)
I am looking for the following article:
Al-Hussaini, A. N. A projective limit view of $L_1$-bounded martingales.
Rev. Roumaine Math. Pures Appl.26 (1981), no.1, 51–54, but I can't find it anywhere.
Do ...
- 573
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
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
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
0
answers
108
views
Has there been any study of the "extreme convergence property" for martingales?
Let $(M_n)_{n \geq 1}$ be a uniformly bounded martingale over a probability space $(\Omega,\mathcal{F},\mathbb{P})$. Define the probability measure $\mu$ on $\mathbb{R}^\mathbb{N}$ to be the law of $(...
- 708
3
votes
0
answers
124
views
How can we show that the quadratic covariation of a Hilbert space valued martingale takes values in the space of nonnegative operators?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete filtration of $\mathcal A$
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ ...
- 167
3
votes
0
answers
221
views
In which sense does the quadratic variation depend on the considered filtration?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge 0}$ be a complete right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$
$X$ be an almost surely ...
- 167
3
votes
0
answers
455
views
Hitting time of two dimensional continuous martingale
Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
- 31
3
votes
0
answers
157
views
Pointwise convergence of ergodic averages of unconventional conditional expectations
Let $(X_i,Y_i)_{i\in\mathbb{Z}}$ be a finite-valued stationary process whose $\sigma$-algebra of tail events is trivial. Let $\mathcal{F}_n^m$ be the $\sigma$-algebra generated by $X_n,\dots,X_m$ ($n,...
- 191
3
votes
0
answers
171
views
compactness of a probability set
I have a question about the compactness of a set of martingale measures. Let $\Omega=\mathcal{C}[0,1]$ be the space of continuous functions on $[0,1]$ and $\mathcal{M}_{\Omega}$ be the family of ...
- 1,835
2
votes
1
answer
198
views
Enlargement of filtration
Let $M_t$ be a continuous time real valued martingale, and $\mathcal F_t$ its natural filtration.
Suppose that $\mathcal F_t \setminus \mathcal F_s$ is nonempty for all $t > s$.
Let $\mathcal G$ be ...
- 6,215
2
votes
1
answer
300
views
Reverse martingale convergence theorem in Banach spaces
In section 1.5 of a course given by Gilles Pisier, the author is claiming that in the excerpt below $\operatorname E[\varphi_i\mid\mathcal A_{-n}]\to\operatorname E[\varphi_i\mid\mathcal A_{-\infty}]$ ...
- 167
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 :
$$P\left(\sup_{t\in [0,...
- 245
2
votes
1
answer
274
views
Inequality for increments of $r$th absolute moments of martingales, $1<r<2$
If $Y_n=\sum_{i=1}^n X_i$ is a martingale, where $X_i$ is a martingale difference sequence, $\mathbb{E}[X_n\mid \mathcal{F}_{n-1}]=0$ for all $n$, we know that
$$ \mathbb{E}\big[Y_n^2-Y_{n-1}^2\big]=\...
- 23
2
votes
1
answer
148
views
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 ...
- 6,215
2
votes
1
answer
167
views
Expected time for a submartingale increasing from A to B
Given $B>A>0$ and $C>0$. Let $\{X_t\}_{t=0}^{\infty}$ be a submartingale with $X_0=A$ and
\begin{equation}
\mathbb{E}[X_{t+1} | \mathcal{F}_t] \geq X_t + C.
\end{equation}
Let $ \tau := \...
- 185
2
votes
1
answer
182
views
Mean of log-normal variable when exponent is replaced by runnung maximum of Ito-integral
Let $W=\{W_t\}_{t\in[0;1]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;1]}$ the filtration generated by $W$, augmented with the nullsets. Let $\{\sigma_t\}_{t\in[0;1]}$ be a continuous and ...
- 335
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
2
votes
1
answer
571
views
Extension of Dynkin's formula, conclude that process is a martingale
This question was asked here, but it did not get enough attention, so I'm crossposting it to MO.
Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial ...
user61522
2
votes
1
answer
74
views
Conditions for absorption
Let $X$ be a Markov chain with countable state space $S$ and transition kernel $P$, and let $h \colon S \to [0,1]$ be a sub-harmonic or super-harmonic function. Assume that for all $\varepsilon >0$ ...
user514580
2
votes
1
answer
2k
views
Alternate proof of Levy’s characterisation of Brownian motion
Levy’s characterisation theorem for Brownian motion states that for a local martingale $X$ with $X_0 = 0$, $X$ is a Brownian motion if and only if it has quadratic variation $\langle X, X \rangle_t = ...
- 6,215
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
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
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
2
votes
2
answers
736
views
Submartingales bounded in $L^p$, $p>1$
Let $p>1$ be a real number. It is known that if $(X_n)_{n\geq 0}$ is a martingale bounded in $L^p$ (i.e. $\sup\{\mathbb{E}(|X_n|^p), n\geq 0\} < +\infty$ ), then $(X_n)_{n\geq 0}$ converges a....
user102214
2
votes
1
answer
238
views
On lower bounds for harmonic functions on $\mathbb{Z}^d$
Consider a non-constant harmonic function $f$ on $\mathbb{Z}^d$ (meaning this that $f(x)$ if the average of the $2d$ values $f(y)$ such that the distance between $x$ and $y$ is one). Let $M_n$ denote ...
- 153
2
votes
1
answer
381
views
Showing convergence in probability of martingale with bounded increments
I am reading a paper which uses the following and I am struggling to show it.
We let $M_n$ be a martingale with bounded increments, wrt the natural filtration $\mathcal{F}_n$. Suppose $M_0=0$. Let $...
- 21
2
votes
2
answers
291
views
A question about Skorokhod embedding problem
The Skorokhod Embedding Problem is well known and has many solutions. Now let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be an embedding to the centered distribution $\mu$, i.e. the ...
- 1,835
2
votes
0
answers
71
views
Assumptions Wald's second equation?
Let $(X_n)_{n\in \mathbb{N}}$ be an i.i.d. sequence of random variables and $N$ an $\mathbb{N}_0$ valued random variable. Let $X_1 \in \mathcal{L}^2$ and $N \in \mathcal{L}^1$. Let $S_n := \sum_{i=1}^...
- 261
2
votes
0
answers
61
views
Characterisation of Bessel process
Let $\delta \in (0, 2)$; $(X_t)_{t \ge 0}$ a nonnegative continuous Markov process. Suppose that
For each $T \ge 0$, if we write $\tau \overset{\mathrm{def}}= \inf\{t \ge T : X_t = 0\}$, then $(X_{T +...
- 177
2
votes
1
answer
246
views
Can we construct close martingales if their terminal marginal laws are close?
Let $M=(M_t)_{0\le t\le 1}$ be a real-valued continuous martingale. Let $\mu := {\rm Law}(M_1)$ and $\varepsilon \in (0,1)$. For any $\nu$ satisfying $W_2(\mu,\nu)\le \varepsilon$, can we construct ...
- 1,399
2
votes
0
answers
121
views
Martingale regularization
Consider a submartingale $X,$ then for almost every $\omega \in \Omega,$ for every $v \in \mathbb{R},\lim_{u \in \mathbb{{Q},u \uparrow v}}X_u(\omega)$ exist in $\mathbb{R}.$
I was wondering if there ...
- 573
2
votes
0
answers
282
views
Identify two continuous martingales in law as time-changed Brownian motions
Let $W$ be a Brownian motion and $\alpha$ be a progressively measurable process taking values in $\mathbb R_+$. Set $\beta_t:=\max(\alpha_t, 1)$ for all $t\ge 0$. Define respectively $X$, $Y$ by
$$X_t:...
- 1,399
2
votes
0
answers
121
views
An unnatural martingale
What is an example of a real valued stochastic process $X$, and a filtration $\mathcal F_t$ such that $X$ is a martingale with respect to $\mathcal F_t$ but not it’s natural filtration?
Either ...
- 6,215
2
votes
0
answers
237
views
Semimartingale decomposition and filtrations
In short: I am trying to understand how the decomposition of a semimartingale into its local martingale and finite variation components depends on the filtration we are using.
So, taking a toy example,...
- 341
2
votes
0
answers
203
views
Is martingale solution equivalent to weak solution for SDE driven by stable process
Consider the following SDE
$$
d X_t=b(X_t)d t+d L_t,
$$
where $L_t$ is the symmetric $\alpha$-stable process. The corresponding generator is given by
$$
L=\Delta^{\alpha/2}+b\cdot\nabla.
$$
Is the ...
- 411
2
votes
0
answers
227
views
Non-negative martingale transforms and Radon Nikodym derivatives
Consider a filtered probability space $(\Omega, (\mathcal F_n), \mathcal F, \mathbb P)$, where $\Omega$ is the set of sequences with value in some $E \subseteq \mathbb R^d$, and $\mathcal F$ is the ...
- 341
2
votes
0
answers
110
views
Modified Pólya's Urn Process
Suppose that we have an urn that initially contains $n$ balls, partitioned into $k\geq 2$ color-classes with respect to some initial probability distribution $P=(p_1,\dots,p_k)$.
At each discrete time ...
- 396
2
votes
0
answers
440
views
Hitting time of a specific Markov chain using martingale approach (or otherwise)
Let $0 < c < 1$. Consider the Markov chain $(X_i)$ on $\{0, 1, \dots, n\}$, with transition probabilities
$$ P(k,k+1) = \left(1 - \tfrac {k}{n} \right)(1-c), \quad k = 0, \dots, n-1, $$
$$ P(k,...
- 985
2
votes
0
answers
130
views
Quadratic characteristic and constancy
Consider a change of measure on $\mathcal{F}_{t}$ defined by the restriction of two probability measures of the form
\begin{align}
\frac{dQ_{t}(\theta)}{dP_{t}}=\exp^{ \theta A_{t}-\kappa(\theta) S_{t}...
- 257
2
votes
0
answers
83
views
Modify Process to a Semimartingale
The original post is from mathstackexchange
According to some difficulties, i decided to ask here again.
Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a ...
- 257
2
votes
0
answers
227
views
Strong law of large number for semimartingale
I just want to know if for semimartingale $X$ we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{\langle X\rangle_{t}}=0$ or when it is possible. I know it is true for Brownian motion.
Thanks
- 31