# Questions tagged [martingales]

The martingales tag has no usage guidance.

275
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### 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 ...

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1
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### 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, ...

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### Construction of a "Dirac" jump process

We work on a probability space $(\Omega,\mathscr{F},\mathbb{P})$ endowed with a filtration $\mathbb{F}$, and consider the positive line $[0,\infty)$. I am wondering if one can make sense of the ...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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 ...

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### 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 $[0,\lambda]$-...

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### 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)...

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### 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 ...

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### 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 ...

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### 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 ...

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### On almost sure convergence of conditional martingales

Let $X$ be a stochastic process with natural filtration $\mathcal F_t$, and $\mathcal G$ a sigma-algebra. Suppose that $X$ is a conditional martingale relative to $\mathcal G$, in the sense that for ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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1
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\varepsilon >0$ ...

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### 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 ...

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### 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 ...

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### 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}...

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### 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 ...

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### 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 $f\in C^2$...

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### 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\{...

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### 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:...

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### 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 $...

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1
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### 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 ...

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### 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
...

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### 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 $\tau_a$ = $\...

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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 $E$ ...

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### 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 $x\in[0,1]$...

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### 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 $\...

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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### Is a Riccati BSDE explicitly solvable?

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...

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### 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 = ...

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### Harmonic function and Markov chain

Let $X=(X_k)_{k \in \mathbb{N}}$ be a Markov chain with countable countable state space $S$ and transition matrix $P.$
Let $\mathcal{T}$ be the tail $\sigma$-field of $X:\mathcal{T}=\bigcap_{k \in \...

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### Proof of yet another extension of deterministic variant of "(Almost) Supermartingale" convergence theorem

In this question, there is a proof for deterministic version of "Almost Supermartingale"
Question: Can we extend [1] as following? If yes, can we prove it?
Let the non-negative sequences be ...

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### A martingale convergence theorem

Let $X$ be a continuous time stochastic process, and denote by $\mathcal F_t$ its natural filtration. We define $\mathcal F_z = \mathcal F_0$ for all $z \leq 0$.
$X$ is said to be strongly predictable ...

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### Proof of extended version of non-random "almost supermartingale"

In this question, a non-random version of "almost supermartingale" theorem is proved.
Here, I would like to extend/apply the non-random version to the slightly different situation. I wonder ...