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Questions tagged [martingales]

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25 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 ...
3
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2answers
103 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 $S_n=X_1+\dotsb+X_n$, ...
1
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1answer
71 views

Martingale representation theorem for symmetric random walk

Let $X(t)$ be a martingale w.r.t. filtration generated by Brownian motion $B(t)$. There is a well-known theorem that states that there is a unique adapted process $H(t)$ such that $$ X(t) = \int_0^t ...
3
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1answer
160 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)...
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1answer
54 views

n-factor martingale representation theorem

Baxter & Rennie at pag. 162 state the following theorem. Let $W$ be an $n$-dimensional $\mathbb Q$-Brownian motion and let $M_t=(M_1(t),...,M_n(t))$ be an $n$-dimensional $\mathbb Q$-martingale ...
3
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0answers
63 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 ...
3
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1answer
66 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 ...
3
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1answer
188 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 ...
7
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1answer
325 views

Is this a martingale sequence?

I have a sequence of random variables $X_1, X_2, \ldots X_N$ such that $|X_i| \leq R \ \forall \ i $, satisfying $$|E[X_n|X_1,X_2,\ldots X_{n-1}]| \leq |X_{n-1}|, $$ Can I construct a sub/super-...
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0answers
53 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 ...
4
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0answers
76 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 $...
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0answers
40 views

Martingale covariation operator in infinite-dimensions

Let $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$ be a filtered probability space $U,H$ be separable $\mathbb R$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ and $(f_n)_{n\in\mathbb N}$...
4
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1answer
139 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 $(\...
1
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1answer
120 views

existence of solution to a martingale optimal transport type problem

I encounter the following problem during the course of my research: Given a random variable $Y=(Y_1,Y_2)$ with values in $\mathbb R^2$ and the cost function $c(x,y)=(x_1-y_1)(x_2-y_2)$ where $x=(x_1,...
2
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1answer
182 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 $i$...
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0answers
155 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 ...
6
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1answer
142 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
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0answers
112 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 ...
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1answer
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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 ...
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0answers
82 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 $(...
2
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0answers
102 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 ...
6
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1answer
310 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. ...
0
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1answer
57 views

Is there any analogous to Levy characterization theorem for purely discontinuous martingales?

Let $M_t$ and $N_t$ be two purely discontinuous martingales such that $[M]_t=[N]_t $ almost surely. Can one conclude that $M$ and $N$ have the same law?
2
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0answers
73 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 ...
1
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1answer
126 views

Optional stopping with unbounded stopping times $\sigma\le \tau$ case

Let $M_t$ be a càdlàg martingale process. Then it is evident, by the optional stopping theorem, that for $\mathcal F_t$-stopping times $\sigma, \tau$ (not necessarily bounded) where $\sigma\le \tau$ ...
2
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1answer
75 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 := \...
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55 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 $(e_n)_{n\in\mathbb N}$ ...
2
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1answer
142 views

Generalisation of Strassen's (Kellerer's) Theorem

Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^d$ with finite first movements, i.e. $$\int_{\mathbb R^d}|x|~\mu(dx),\quad \int_{\mathbb R^d}|x|~\nu(dx) \quad <\quad +\infty.$$ $\mu$...
3
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1answer
134 views

Concentration inequality of joint event over time of a submartingale

Consider a discrete time submartingale $X_n$ with bounded difference $|X_n-X_{n-1}|\leq c$. With Azuma inequality we have the concentration of a single time event as $$ P(X_t-X_0 \leq -t) \leq exp\...
6
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1answer
339 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 $W$ ...
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4answers
639 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 ${\cal F}_n$-...
4
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0answers
209 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 ...
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84 views

Using the optional stopping theorem on a stochastic process

(I'm much more used to number theory than to stochastic processes, so there are probably a lot of errors in the following:) Consider a stochastic differential equation $dx = F(t,x) dt + \sigma dW$, ...
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0answers
57 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 ...
4
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1answer
223 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 ...
4
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1answer
261 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 ...
6
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1answer
232 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 ...
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0answers
218 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,...
5
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1answer
89 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 | \...
2
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1answer
318 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....
2
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1answer
135 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 ...
2
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1answer
119 views

martingale convergences wrt dyadic intervals

Consider $[0,1]$ and thereon $D_n = \{ [k2^{-n}, (k{+}1) 2^{-n} ) : 0\le k \le 2^n-1\}$ and $\mathcal F_n$ the generated sigma-algebras by $D_n$. We do know that uniformly bounded (in $L_p$-norm) ...
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1answer
68 views

Approximate martingales by truncation

Let $(X,Y)$ be a $\mathbb R-$valued martingale. For any $\varepsilon>0$, is it possible to find another martingale $(X',Y')$ s.t. $X'$ and $Y'$ are supported on a compact set, and $$ \mathbb E\big[\...
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0answers
85 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 ...
7
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2answers
196 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 ...
2
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1answer
200 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 $...
6
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0answers
130 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 $$...
12
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1answer
238 views

Convergence of an implicitly defined sequence of random variables

Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
2
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0answers
65 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}...
0
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1answer
319 views

Predictable quadratic Variation <.> has same intervals of constancy as the process

From Revuz and Yor - Continuous Martingales and Brownian Motion 1999 Chapter IV Proposition 1.13 it is proven, that for a continuous local martingale $M_t$ the intervals of constancy ...