All Questions
Tagged with pr.probability martingales
210 questions
7
votes
1
answer
560
views
Doob's inequality for martingale "convolution"
Let $(X_t, t \in \mathbb{N})$ be a martingale, and let $a \leq b \leq T \in \mathbb{N}$ be constants. Is there something like Doob's inequality for $\mathbb{E} \sup_{a \leq t \leq b} X_t(X_T-X_t)$, i....
- 3,958
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 ...
- 29.8k
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=...
- 4,746
10
votes
1
answer
2k
views
Law of large numbers for martingales
I apologize in advance if this question is too basic, but I've received no response on Math Stack Exchange, so perhaps it is more appropriate here:
Let $X_n$ be a square-integrable martingale with $\...
- 63.9k
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 ...
CommunityBot
- 1
5
votes
1
answer
652
views
Proof of Pinelis (1992) - Banach space inequalities
I am reading Pinelis "An approach to inequalities for the distributions of infinite -dimensional martingales" and cannot follow his proof of Theorem 3:
Let $(f_n)$ be a martingale in a separable ...
- 128k
1
vote
2
answers
3k
views
Is stopped brownian motion not a martingale?
In page 45 of the book "Financial Derivatives In Theory and Practice by P.J.Hunt and J.E.Kennedy, it seems to me that the author says the stopped Brownian Motion is not a martingale as follows.
(...
- 193
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
$$
\mathbb{E}\left[
\int_0^{\...
- 63.9k
2
votes
1
answer
287
views
Bernstein Inequality for continous local martingale
I'm looking for a simple proof of the following fact, which is somehow Bernstein inequality in continuous time.
Let $(M_t)_{t\geq 0}$ be a continuous local martingale. Then :
$$P\left(\sup_{t\in [0,...
- 3,958
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
4
votes
1
answer
456
views
Weaker version of the martingale convergence theorem
Let $\mathcal{A}_n$ be a sequence of finite sigma-algebras, let $\mathcal{B}_{q,p}= \sigma(\mathcal{A}_n, q \geq n \geq p )$. Moreover, we suppose $\mathcal{A}_k \subset \mathcal{B}_{\infty,p}$ for ...
- 3,486
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 ...
- 17.2k
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 $\...
- 984
7
votes
1
answer
975
views
Prove an anti-concentration inequality for a martingale
My problem can be described easily:
I have a sequence $(X_l)_{l \in \mathbb{N}}$ of r.v. adapted to some filtration $(\mathcal{F}_l)_{l \in \mathbb{N}}$, such that
$\left|X_{l+1}-X_l\right|\le R$ a. ...
- 63.9k
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 ...
- 63.9k
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 $\|...
- 10.4k
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 $S_n=X_1+\dotsb+X_n$, ...
- 173
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)...
- 3,669
3
votes
1
answer
139
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
...
- 128k
3
votes
1
answer
235
views
Inequality for exponential sum in Dvoretzky 1972
I'm currently trying to figure out the following inequality. It looks like an inequality for the exponential sum, but I can't verify it or find a source explaining it any further. Most likely it has ...
- 128k
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
4
votes
0
answers
238
views
Does Novikov condition imply BMO martingale?
Let $(\Omega,\mathbb{F},P)$ be a complete probability space, equipped with a filtration $\mathcal{F}_t, 0 \le t < \infty$. Consider a continuous local martingale $(X_t, \mathcal{F}_t)$ such that $...
- 195
4
votes
1
answer
302
views
Zero-one law for an independence-like structure
I am a number theorist by profession, so apologies if the answer to this question is "trivially true" or "trivially false".
Let $(\Omega, \mathcal{A}, P)$ be a (non-atomic) probability space. Let $(\...
- 29.8k
6
votes
1
answer
653
views
Change of space-time in Walsh's stochastic integral
One can read about Walsh's construction of martingale integral in the paper (pp.16-23)
http://www.math.utah.edu/~davar/ps-pdf-files/SPDEBookDK.pdf (Wayback Machine)
For $U,V\in \mathcal{B}(\mathbb{R}\...
- 4,713
4
votes
1
answer
1k
views
Does variants of Bernstein and Freedman concentration inequalities exist with NO uniform bound on the range of RV or martingale differences
A classic formulation of the Bernstein inequality (from Wikipedia) is as follow:
Let $X_1, \ldots, X_n$ be independent zero-mean random variables. Suppose that $|X_i|\leq M$ almost surely, for all $i$...
- 128k
8
votes
3
answers
2k
views
What is the optimal growth of the constant in BDG?
Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} |X|^...
- 3,946
7
votes
1
answer
466
views
Martingale version of Bernstein-type inequality for (slightly) heavy-tailed distributions?
It is known that for sub-exponentially distributed martingale difference sequence, the following Bernstein-type inequality holds:
$$
ℙ\left(\left|
\sum_{i=1}^N a_i X_i
\right| \ge t \right)
\le
2\...
- 4,713
5
votes
0
answers
543
views
Vector martingale concentration
Let $\varepsilon_1, \dots, \varepsilon_N$ be a martingale difference sequence in $R^d$ with $\|\varepsilon_n\| \le B_n, a.s.$ for each $n=1,\dots,N$. Do we have some Azuma-type concentration ...
- 719
-2
votes
1
answer
113
views
Demonstrations on an $L^1$ martingale [closed]
If $(X_n,\mathcal{F_n})_{n\in \mathbb{N}}$ is a martingale such that $\forall$ n $\in \mathbb{N}, \frac{X_{n+1}}{X_n}\in L^1$ How can be demonstrated that:
$\mathbb{E}[\frac{X_{n+1}}{X_n}]=1$ and ...
- 9
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
5
votes
4
answers
1k
views
Examples of discrete time martingales
In probability, a martingale is given by a sequence of integrable
random variables $(S_n)$ and an increasing sequence of
$\sigma$-algebras ${\cal F}_n$ such that
$S_n$ is ${\cal F}_n$-...
- 47.4k
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
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 := \...
- 29.8k
6
votes
0
answers
183
views
Distribution of the stopping time of an autoregressive sequence
Consider $e_t$ being i.i.d. uniformly chosen from $\pm 1$. Let $\eta$ be a small positive constant. What is the distribution of $T$ such that $\eta^{0.5} (1+\eta)^T W_T$ first hits $\pm 1$, in which
$$...
- 1,127
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
5
votes
1
answer
123
views
Convergence of conditional second moments
Let $(\Omega, \mathcal{A},P)$ be a probability space, and let $(\mathcal{F}_k)_{k \geq 1}$ be a filtration which converges to $\mathcal{A}$. I suppose it is true that
$$
E \left( \big(E \left( X | \...
- 3,958
6
votes
3
answers
2k
views
Iterated Ito Integral, Gaussian Volterra Process
Let me define
$$
J^f_{n}(t) = \, \int_0^t \int_0^{t_1} \ldots \int_0^{t_{n-1}} f(t, t_1, \ldots, t_n) \; dB_{t_n} ...dB_{t_1}
$$
where $f:[0,1]^{n+1} \to \mathbb{R}$ is a nice deterministic function....
- 7,514
7
votes
2
answers
594
views
Large deviation/concentration inequality for submartingale
Let $S_t = M_t + D_t$ be the sum of a martingale $\left(M_t\right)_{t=1,2,\ldots}$ and a predictable process $(D_t)_{t=1,2,\ldots}$ such that the variance of the increments of $M$ is uniformly bounded ...
- 3,958
7
votes
1
answer
1k
views
Moment bounds on exponential martingale
Consider the exponential martingale used in the Girsanov transformation of
measure:
$$Z(t) = \exp\Big(\int_0^tXdW - \frac{1}{2}\int_0^t|X|^2ds\Big)$$
so that $Z$ solves the sde $dZ = ZXdW$ where $W$ ...
- 984
5
votes
0
answers
653
views
Explicit martingale representation for a Brownian bridge
Let $W$ denote a Wiener process, $\displaystyle M_t = \max_{0 \le s \le t} W_s$ its running maximum. The martingale representation of $M$ is known explicitly:
$$M_T = \sqrt{\frac{2T} \pi} + \int_0^T ...
- 341
-1
votes
1
answer
519
views
Poisson kernel is the Cauchy distribution, reference?
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?
CommunityBot
- 1
0
votes
0
answers
65
views
Wanted: example of a non-stationary sequence with reverse empirical measure
Assume we have a sequence $\xi=(\xi_1,\xi_2,\dots)$ of random variables such that $$\eta=\left(\frac{\sum_{i=1}^n \delta_{\xi_i}}{n}\right)_{n\geq 1}$$ is a reverse-martingale with respect to its own ...
- 211
4
votes
1
answer
443
views
Uniform martingale convergence of Radon-Nikodym derivatives of a convex set of probabilities
Cross posted at MSE here. I'm hoping someone here can help complete zhoraster's answer. Any hints or references are appreciated.
Let $(\Omega, \mathcal{F})$ be a measurable space equipped with a ...
CommunityBot
- 1
5
votes
1
answer
1k
views
Supremum of a martingale
Let $(X_n)$ be a martingale. What can be said about the distribution of its maximum over a window of fixed length:
$$M_n = \max_{n-10 \leq k \leq n} X_k$$ or about the "range" over a window:
$$R_n = \...
CommunityBot
- 1
6
votes
1
answer
956
views
History of optional sampling/stopping theorem
Does anyone have a good explanation of the name, and why Doob chose it? It states the following: if $T$ is a stopping time such that $\mathbb{P}(T < \infty)$, and $M_n$ is a uniformly integrable ...
- 188k
6
votes
1
answer
461
views
Gronwall lemma with conditional expectation
The discrete Gronwall's inequality states that if $x_n$ and and $u_n$ are non-negative sequences such that
$$ x_{n+1}\le a+\sum_{k=0}^n u_k x_k$$
then $$x_n\le a\prod_{k=0}^{n-1} (1+u_k)$$
(It can be ...
CommunityBot
- 1