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24 views

Characterisation of a family of continuous martingales

I look for a full characterisation of the continuous martingales $X=(X_t)_{0\leq t\leq T}$ (defined on some filtered probability space as nice as possible) such that $$X_0=0\quad \mbox{ and } \quad\...
0 votes
0 answers
31 views

Looking for a citation for this simple generalization of the Markov bound to non-negative super-martingales

Does anybody know a reference for the following theorem? Theorem 1. Let $(X_t)_{t=0}^\infty$ be a non-negative supermartingale. Then, for any constant $c > 0$, the event $(\exists > t)\, X_t \...
4 votes
1 answer
66 views

Expectation bounds on supremum of family of martingales

Suppose I fix a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, P)$ and on it a Brownian motion $B$. Let $\tau_\alpha$ denote a set of stopping times which satisfies $\sup_\alpha \tau_\...
0 votes
1 answer
57 views

Lower bounding an alternating series with signs from a martingale difference sequence

Let $\epsilon_n \in \{-1, 1\}$ be a martingale difference sequence, in the sense that $$M_n := \sum_{i = 0}^n \epsilon_i$$ is a martingale. We assume $\epsilon_0 = \pm 1$ with probability $\frac{1}{2}$...
6 votes
1 answer
660 views

On the martingale betting scheme

For a fixed probability $0 < p < 1$, let $X^p$ be the martingale that goes up by $1$ with probability $p$, and goes down by $\frac{p}{q}$ with probability $q := 1-p$. Write $X$ for the ...
3 votes
1 answer
181 views

A nice terminal inequality for martingales

Let $X_t$ be a continuous time martingale taking with $\sup_t \mathbb E[X_t^-] < \infty$, and $X_0 = 0$ almost surely. Assume further that $X_1$ admits a probability density function. Is it true ...
1 vote
1 answer
60 views

Reverse Doob’s maximal inequality for bounded martingales

Consider the set of discrete or continuous time $L^\infty$-bounded martingales $X$ with $X_0 = 0$ almost surely. Here $L^\infty$-bounded means $\|X\|_{\infty} := \sup_t \mathbb \|X_t\|_{L^\infty(\...
0 votes
1 answer
64 views

Sharpening Doob’s upcrossing inequality for Brownian motion

Note: This question is heavily related to a series of posts ([1], [2]) by user GJC20. Provided a martingale $X$ in continuous-time, Doob's upcrosssing inequality states: If $U(a,b)$ denotes the number ...
0 votes
2 answers
60 views

Do continuous martingales satisfy this nice terminal inequality?

Let $X$ be a continuous, non negative martingale on $[0, 1]$ with $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. Assume further that $X_1$ admits a probability density function. Is it true that the ...
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}^...
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 vote
1 answer
185 views

Sum of $X_k$ with $\mathbb{P}(X_k=\pm 1) = 1/2\pm 1/(2\sqrt{k})$

Let $\{X_k\}$ be a sequence of mutually independent random variables with \begin{align} \mathbb{P}(X_k = 1) & = \frac{1}{2} + \frac{1}{2\sqrt{k}}, \\ \mathbb{P}(X_k = -1) & = \frac{1}{2} - \...
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)...
9 votes
3 answers
448 views

All stationary martingales are constant?

Suppose $(X_{n})_{n\geq{1}}$ is a stationary process that is a martingale with respect to some filtration. Suppose also that $\mathbb{E}X_{0}^{2}<\infty$ so that $\mathbb{E}X_{n}^{2}<\infty$ for ...
1 vote
1 answer
464 views

A Lévy process is a semimartingale proof

I have to prove that a Lévy process is a semimartingale. In general we say that $X$ is a semimartingale if it is an adapted process such that, for each $t ≥ 0$, $$X (t) = X (0) + M(t) + C(t)$$ where $...
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 +...
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 ...
1 vote
1 answer
129 views

A martingale puzzle about sum of expected squared bounds

I'm trying to get one of those "with $1-\delta$ probability, the following holds"-style bounds, and the following martingale problem looks solvable by some Freedman or Bernstein-style bound, ...
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 ...
10 votes
3 answers
4k views

Extension of the Azuma-Hoeffding inequality (when the differences are bounded with large probability)

Let $(X_i)$ be a super-martingale and suppose their differences are bounded ''with high probability'', that is $$\mathbb{P}(\exists\,i=1,\dots,n\text{ s.t. }|X_i-X_{i-1}|>c_i) \,\leq\, \epsilon$$ ...
4 votes
0 answers
80 views

Does this filtration have a name?

In the context of Ethier&Kurtz Markov Processes: Characterization and Convergence (Chapter 4, equation (3.2)) as well as the two papers Martingale problems for conditional distributions of Markov ...
4 votes
1 answer
3k views

The only continuous martingales with stationary increments are Brownian motions

I know that the above statement is true, but I can't demonstrate it. It's a pretty powerful theorem, here is its mathematical formulation: Theorem: The only continuous martingales with stationary ...
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 $\...
0 votes
0 answers
90 views

Martingale defined by an integral

Consider a probability space $(\Omega,\mathcal{F},P).$ Let $f \in C^{\infty}_{c}(\mathbb{R}^d,\mathbb{R}),p \geq 2.$ $(X_r^{y})_{(r,y) \in \mathbb{R}_+ \times \mathbb{R}^d}$ is a stochastic process ...
6 votes
1 answer
168 views

Weak convergence of random measures generated by non-negative martingales?

If I have a sequence of non-negative continuous martingales $(M_n(x))_{n\ge 1}$ on $x\in[0,1]$, i.e. for each fixed $n$, $M_n:[0,1]\to[0,\infty)$ is a continuous process, and for each fixed $x\in[0,1]$...
5 votes
2 answers
642 views

Examples of a continuous martingale with $E[\sup\limits_{0\leq s\leq t} |M_s|]=\infty$?

A local martingale is a martingale iff it is in the class DL. The condition: for every $t\in[0,\infty)$ $$E[\sup\limits_{0\leq s\leq t} |M_s|]<\infty\tag1$$ guarantees a local martingale $M$ is ...
9 votes
1 answer
4k views

Quadratic variation and predictable quadratic variation for martingales

Let $(M_{t})_{0\le t\le 1}$ be a continuous martingale with respect to the filtration $(\mathcal{F}_{t})_{0\le t\le 1}$. Assume that $E M_1^2<\infty$. Fix $N$ and consider now a discrete version ...
4 votes
5 answers
2k views

Martingales and Betting Strategies

Does anyone know of a good introduction to the theory of martingales and betting strategies from the point of view of statistics and/or probability theory? I'm looking for something basic, with lots ...
1 vote
0 answers
125 views

Can we construct close discrete martingales if their terminal marginal laws are close?

As no answer or comment to Can we construct close martingales if their terminal marginal laws are close? we consider a simplified version (discrete-time) as below: Let $M=(M_k)_{0\le k\le n}$ be a ...
6 votes
1 answer
396 views

Is a martingale conditioned to be large a submartingale?

Let $X$ be a continuous time martingale such that $X_\infty := \lim_{t \to \infty} X_t$ exists almost surely. Let $x \in \mathbb R$ be such that $\mathbb P(X_\infty \geq x) > 0$, and define the ...
10 votes
2 answers
828 views

On martingale convergence

Let $(X_t)_{t\ge0}$ be a martingale with continuous paths. It was previously shown here and here that then it is impossible that $X_t\to\infty$ almost surely as $t\to\infty$. Is it possible that there ...
4 votes
2 answers
373 views

Another curious martingale

This is a natural follow up question to A curious martingale. Does there exist an almost surely continuous martingale that converges in probability to $+\infty$? Note: We say a process $X_t$ converges ...
7 votes
2 answers
2k views

A curious martingale

Does there exist an almost surely continuous martingale $X$ with $X_t \to +\infty$ almost surely? Remark: Note that such a martingale exists in discrete time, or equivalently in continuous time if the ...
11 votes
2 answers
2k views

De Finetti's theorem, the pointwise ergodic theorem, and reverse martingales

De Finetti's theorem says that an exchangeable sequence of random variables $X_i$ is a mixture of i.i.d. random variables. In other words, if $\mu$ is a measure on $\mathbb{R}^\infty$ that is ...
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 ...
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 $\varepsilon >0$ ...
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 ...
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 ...
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 ...
1 vote
1 answer
83 views

Integral of $M^\text{*} - M$ with respect to $M^\text{*}$ is zero for $M^\text{*}$ the running maximum of $M$ a continuous local martingale

Given $M$ a continuous local martingale, and $M^\text{*} = \sup_{0 \leq s \leq t} M_s$ its running maximum, we consider the finite variation integral $$ I_T:= \int_0^T (M^\text{*}_s - M_s) \, \text{d}...
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 vote
0 answers
156 views

is there a discrete version of Dambis Dubins Schwarz Theorem

Theorem (Dambis, Dubins-Schwarz). If $M$ is a $\left(\mathscr{F}_t, P\right)$-continuous martingale vanishing at 0 and such that $\langle M, M\rangle_{\infty}=\infty$ and if we set $$ T_t=\inf \left\{...
3 votes
1 answer
352 views

First time random sum exceeds value

Suppose $X_n$ $n = 1, 2, \ldots$ are i.i.d random variables with $\mu := \mathbb{E}[X_n]$ > 0. (although they are not necessarily non-negative). Then if $S_n = \sum_{k=1}^n X_k$ and $\tau_a$ = $\...
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 ...
1 vote
0 answers
87 views

Normal approximation of martingale difference

Apologies in advance if the question is not precise (or silly), I am not a probabilist by profession. I have the following question: Let $(X_n)_{n \geq 1}$ be a martingale difference sequence. Assume ...
8 votes
1 answer
533 views

Concentration bounds for martingales with adaptive Gaussian steps

Consider the following martingale: $X_1 \sim \mathcal{N}(0, 1)$, and for any $n > 1$, $X_n \sim \mathcal{N}(X_{n-1}, X_{n-1}^2)$ (notice, this is a conditional distribution given $X_{n-1}$). I am ...
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 = ...
5 votes
1 answer
350 views

Can an a.s. non constant continuous martingale be differentiable with nonzero probability?

Let $M$ be a continuous martingale such that almost surely, the sample paths of $M$ are not constant. Question: Is it true that $M$ is almost surely not differentiable?
1 vote
0 answers
182 views

Hardy's inequality proof using Doob's inequalities

Consider a probability space $([0,1],\mathcal{B}([0,1],\lambda),p>1$ and $f \in L^p(]0,\infty[).$ We want to prove Hardy's inequality using martingale theory and Doob's maximal inequalities. Let $\...
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 ...

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