Questions tagged [martingales]

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3 votes
1 answer
78 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 $\...
2 votes
0 answers
46 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 ...
0 votes
0 answers
80 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
128 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 $x\in[0,1]$...
2 votes
0 answers
61 views

Upcrossing lemma and subharmonic functions

I have been studying the upcrossing lemma for submartingales, which asserts that if $X_n$ is a non negative submartingale, and $ \lambda>0$ then if we denote by $U_n$ the number of $[0,\lambda]$-...
1 vote
0 answers
80 views

Gluing theorem for martingales

Let $M=(M_t)_{1\le t\le 2}$ be a continuous (resp. right-continuous) martingale. Denote $x:=\mathbb E[M_1]\in\mathbb R$. Can we construct on some probability space a continuous (resp. right-continuous)...
0 votes
0 answers
17 views

Construct continuous martingales that are close to constants

Let $\mu_0,\mu_1$ be probability measures on $\mathbb R$ that are of finite second moment and increasing in convex order, i.e. $$\int_\mathbb R f(x)\mu_0(dx) \le \int_\mathbb R f(x)\mu_1(dx)$$ holds ...
2 votes
1 answer
181 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 ...
3 votes
2 answers
176 views

Can any right-continuous martingale be approximated by continuous ones?

It is known that any function that is right-continuous with left limits (càdlàg as a French abbreviation) can be approximated by continuous ones (under e.g. Skorokhod topology). Let $M=(M_t:0\le t\le ...
5 votes
2 answers
610 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 ...
3 votes
1 answer
290 views

Trajectory regularity of conditional expectation with additional randomness

Consider a probability space that support a standard Brownian motion $W=(W_t)$ and a random variable $Z$ that is independent of $W$. Denote by $\mathbb F^W=(\mathcal F^W_t)_t$ the natural filtration ...
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 ...
6 votes
0 answers
155 views

On almost sure convergence of conditional martingales

Let $X$ be a stochastic process with natural filtration $\mathcal F_t$, and $\mathcal G_t$ another filtration. Suppose that $X$ is a conditional martingale relative to $\mathcal G_t$, in the sense ...
1 vote
0 answers
123 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 ...
0 votes
1 answer
63 views

Martingale property and martingale property in law

Let $(\Omega,\mathcal F, (\mathcal F_t)_{t \in T}, P)$, $\, T \subseteq \mathbb R$, be a filtered probability space. A stochastic process $X=(X_t)_{t\geq 0}$ adapted to $\mathcal F_t$ is an $\mathcal ...
0 votes
1 answer
278 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 $...
6 votes
1 answer
352 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 ...
5 votes
1 answer
382 views

On the convergence of a martingale

Let $W$ be a standard one dimensional Brownian motion and let $A$ be the process defined by : $$\forall \ t\geq 0: \quad A_t := \int_0^t\left(1 + e^{W_s}\right)\mathrm{d}s$$ and for $t\geq 0$, we ...
9 votes
2 answers
626 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
326 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
1k 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
581 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
72 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
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 ...
3 votes
0 answers
133 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
114 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
80 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}...
4 votes
0 answers
200 views

A notion of SDE via the martingale representation theorem

$\newcommand{\d}{\mathrm{d}}$It is well-known that differentiating stochastic processes with respect to time is usually impossible in the usual sense. For instance, a Brownian motion $W$ on a ...
2 votes
1 answer
130 views

Local martingale with increasing process

Here is a problem in stochastic calculus: If $M_t$ is a continuous process and $A$ an increasing process, then $M$ is a local martingale with increasing process $A$ if and only if, for every $f\in C^2$...
2 votes
0 answers
266 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
107 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\{...
1 vote
1 answer
154 views

On a martingale defined via some SDE

Let $W$ be a one-dimensional Brownian motion. Consider the stochastic differential equation (SDE) $$dX_t = C(t)(1-X_t)dW_t,\quad \forall t\ge 0,$$ where $C$ is a continuous and bounded function. Under ...
0 votes
0 answers
158 views

A variant of Dubins–Schwarz's theorem

Let $W$ be a Brownian motion and $\alpha$, $\beta$ be two progressively measurable processes taking values in $\mathbb R_+$ s.t. $\alpha_t\le \beta_t$ for all $t\ge 0$. Define respectively $X$, $Y$ by ...
3 votes
1 answer
183 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
137 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
76 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
505 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 ...
3 votes
0 answers
67 views

Making a space UMD via interpolation

Recall that a Banach space $B$ has Unconditional Martingale Difference (UMD-$p$) if there is a constant $C_p$ such that for every $B$-valued martingale difference sequences $(d_n)_n$ and choice of $\...
2 votes
1 answer
1k 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
327 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
1 answer
302 views

Does a continuous martingale converge almost surely on the event that its quadratic variation is finite?

Let $M$ be a continuous martingale. Denote by $E$ the event that its total quadratic variation is finite, i.e. $$E := \{\langle M, M \rangle_\infty < \infty\}.$$ Question: Is it true that as $t \to ...
2 votes
1 answer
403 views

Is a martingale constant on the event that its quadratic variation is zero?

Let $M_t$ be a continuous time martingale, and assume its quadratic variation is identically zero with some positive probability less than $1$. To be more precise, assume there exists some event $E$ ...
1 vote
0 answers
172 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
157 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 ...
2 votes
1 answer
153 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 ...
1 vote
1 answer
125 views

Characterization of Brownian motion: processes with right-continuous paths

I am looking for a reference with a proof for the following fact: If a right-continuous martingale $(X_r)_{ r \geq 0}$ is such that $X_0=0,(X^2_r-r)_r,(X_r^3-3rX_r)_r,(X_r^4-6rX_r^2+3r^2)_r$ are ...
2 votes
0 answers
59 views

Martigale that maximizes its expected number of upcrossings/downcrossings

Let $T\ge 1$ be some fixed integer. Consider a discrete-time martingale $(X_t)_{t=0,1,\ldots, T}$ or a continous-time martingale $(X_t)_{0\le t\le T}$ (the latter can be continuous or cadlag if it ...
7 votes
1 answer
447 views

A note on Doob's theorem

I have faced the following problem, regarding to the Martingale Theory. Because this area far from my area I don't know whether this problem is in literature or this can be simple question for ...

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