Questions tagged [martingales]

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Martingale diffusions falling in $\{-1,1\}$ at finite maturity

This is a continuation of Characterization of martingale diffusions ending in $\{-1,1\}$ $X=(X_t)_{0\le t\le T}$ is said to be a martingle diffusion if $X_0=0$, $X_T\in\{-1,1\}$ and $$X_t=\int_0^t a(u,...
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Characterization of martingale diffusions ending in $\{-1,1\}$

Let $\mathcal M$ be the collection of martingle diffusions starting at zero and ending in $\{-1,1\}$. Equivalently, $X\in \mathcal M$ iff there exists a measurable function $a$ s.t. it holds almost ...
GJC20's user avatar
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2 votes
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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 $\operatorname E[\varphi_i\mid\mathcal A_{-n}]\to\operatorname E[\varphi_i\mid\mathcal A_{-\infty}]$ ...
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1 answer
227 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]=\...
mattia's user avatar
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How to quantify the randomness of martingales?

For a real valued random variable (or probability distribution), the (relative) entropy is used to quantify how random it is. Provided a stochastic process, how can we determine whether it is ''very ...
user avatar
2 votes
1 answer
589 views

If a continuous function of a Markov martingale is a martingale, does the function have to be affine linear?

Let $M$ be an almost surely continuous martingale that is not almost surely constant in time - that is, it is not the case that almost surely, $M_t = M_0$ for all $t$. Assume further that $M$ is a ...
Nate River's user avatar
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4 votes
2 answers
233 views

Bounded density for diffusions with diffusion coefficients bounded away from $0$

Consider a diffusion given by $$X_t=\int_0^t a(s,X_s)\,dW_s$$ for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
Iosif Pinelis's user avatar
5 votes
2 answers
270 views

A comparison of diffusions

Consider two diffusions given by $$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$ for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough ...
Iosif Pinelis's user avatar
1 vote
1 answer
190 views

First hitting time for non-homogeneous diffusion martingale

This question can be seen as a continuation of Lipschitz continuity of $\mathbb P[\tau>t]$ with respect to $t$ Consider the martingale given as $$X_t=1+\int_0^t a(s,X_s)dW_s,\quad \forall t\ge 0.$$ ...
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Martingale representation theorem for almost adapted martingales

Given a filtration $\mathcal F_t$ on a probability space, we say a stochastic process $X$ is almost $\mathcal F_t$-adapted if there exists some $\mathcal F_t$-adapted process $Y$ such that $\underset{...
Nate River's user avatar
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2 votes
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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 ...
Nate River's user avatar
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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 $\mathcal H_\infty$-...
Nate River's user avatar
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Probability of filling a small ball before exiting a big one for $d=2$

Let $S_n$ be the simple random walk in dimension $d=2$. Let $0<r<R$ and $\alpha \in (0,1)$. Let $B_r$ denote the $\{x \in \mathbb Z^2: \|x\|\le r\}$ where $\|\cdot\|$ is the Euclidean norm. ...
Kernel's user avatar
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Does this sequence of martingales converge?

Consider a sequence of martingales that are right-continuous with left limits, denoted by $(X^n_t)_{0\le t\le 1}$, such that for each $n\ge 2$, \begin{eqnarray} (1) && X^n_0=0 \mbox{ and } \...
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12 votes
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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 ...
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$\exists c \in\mathbb{R}_+^*,\forall p,r\in \mathbb{R}_+,E[|X_{p+r}-X_r||\mathcal{F}_r] \leq c$ 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}-...
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Martingale representation of a stopped Brownian motion

This question follows from the previous post Question on the martingale representation theorem which has not been answered. I consider thus a particular case. Let $(B_t)_{t\ge 0}$ be a standard ...
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Can a continuous mean zero process be turned into a semimartingale via a change of measure?

Let $X_t$ be a continuous process such that $E[X_t] = 0$ for all t. Denote by $\mathcal F_t$ the completion of its natural filtration. Does there exist some $F_{\infty}$-measurable non negative random ...
Nate River's user avatar
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1 answer
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Question on the limit of martingales

I am looking for the condition/criterion that yields the convergence of right-continuous martingales, motivated by the following question. For $M,N\ge 1$, set $I_M:=\{t_m\equiv m/M: 0\le m\le M\}$ ...
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2 votes
1 answer
503 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 ...
Kurt.W.X's user avatar
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5 votes
3 answers
937 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?
Nate River's user avatar
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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 ...
Joe_Affine's user avatar
1 vote
1 answer
129 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}...
J. Doe's user avatar
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2 votes
1 answer
143 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 ...
Nate River's user avatar
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4 votes
1 answer
588 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 ...
Nate River's user avatar
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1 vote
1 answer
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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 ...
legon's user avatar
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0 answers
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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 $...
Kurt.W.X's user avatar
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3 votes
0 answers
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On the property of a nonnegative stochastic process "attracted" near zero

Let $\{X_k\}$ be a nonnegative stochastic process satisfying $$E\left[ X_{k+1} \mid \mathcal{F}_k \right] \leq \rho X_k + c,$$ where $0 < \rho < 1, c>0$. Intuitively, the process is likely to ...
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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{...
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0 answers
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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 \...
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3 votes
0 answers
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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 ...
Daron's user avatar
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1 vote
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425 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, ...
SMS's user avatar
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1 answer
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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)...
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0 answers
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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 $(X_t)_{t\in[0,1]}$ ...
ABIM's user avatar
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0 answers
586 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 ...
Martin Weizenguss's user avatar
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0 answers
42 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 ...
Hedonist's user avatar
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10 votes
4 answers
637 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). ...
John Wong's user avatar
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2 answers
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Characterization of the generator of a Lévy process using martingale problems

Let $(X_t)_{t\ge0}$ be a real-valued Lévy process. Note that $$\mu_t:=\mathcal L(X_t)\;\;\;\text{for }t\ge0$$ is a continuous convolution semigroup$^1$. Let $$\tau_x:\mathbb R\to\mathbb R\;,\;\;\;y\...
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0 answers
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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,...
Tartrate's user avatar
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3 votes
2 answers
551 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? $$...
Siam's user avatar
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2 answers
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Expectation of Brownian motion increment and exponent of it

While reading a proof of a theorem I stumbled upon the following derivation which I failed to replicate myself. Let $\mu$ be a constant and $B(t)$ be a standard Brownian motion with $t > s$. Show ...
James's user avatar
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1 answer
273 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 ...
James's user avatar
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13 votes
1 answer
685 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 ...
Dor's user avatar
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1 answer
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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 ...
user123124's user avatar
2 votes
1 answer
264 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 ...
Kernel's user avatar
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0 answers
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Exponentially suppressed events for bounded difference super-martingales

Let $\{ Z_n \mid n = 0,1,..\}$ be a non-negative super-martingale and assume that it is of bounded difference i.e $\exists ~c_i >0$ s.t $\vert Z_{i+1} - Z_i \vert \leq c_i$. Then we know (Azuma-...
gradstudent's user avatar
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6 votes
1 answer
363 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 ...
Ever Garden's user avatar
2 votes
0 answers
165 views

Non-integer conditional moment of exponential functional of Brownian motion

Let $B_t$ be a standard Brownian motion. I want to solve the following: $$ \mathbb{E}\left[\left(\int_0^1 e^{\sigma B_t}dt \right)^{1/(1-\beta) }\mid e^{\sigma B_1}=z \right], $$ for some fixed $0<\...
Seung Hyeon Yu's user avatar
0 votes
1 answer
1k 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 ...
Math is like Friday's user avatar
2 votes
1 answer
749 views

Calculate Radon-Nikodym derivative

For the laws of two pure-jump Markov processes $\mu_1$ and $\mu_2$ on $\mathbb R^n$, which generators are $H_1f(x)=\int h(x,dy) (f(y)-f(x))$ and $H_2f(x)=\int e^{-g(x,y)} h(x,dy) (f(y)-f(x))$ (...
Ivan Petrov's user avatar