# Questions tagged [martingales]

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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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### 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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### 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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### 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 ...
1 vote
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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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### 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  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 ...
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### 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?
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### Can we invoke "almost supermartingale" Theorem for deterministic sequences?

Perhaps stupid question. Question: Can "almost supermartingale" theorem be equally applicable to prove the convergence of some algorithms solving non-random optimization problems? Attempt ...
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### Martingales converging in probability but not a.s

It is known that a random series $$\sum_{n\geq 1} X_n$$ whose terms $X_n$ are independent converges a.s. if and only if it converges in probability. Is it true that a martingale $(Y_n)$ converges a....
1 vote
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### Where to submit a new proof of the continuous martingale convergence theorem?

There were various proofs of the discrete martingale convergence theorem, but as far as I know there is only one proof of the continuous version of this theorem using the up-crossing lemma. I wrote a ...
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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 ... 435 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 ...
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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 ...
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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 $\mathcal H_\infty$-...
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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. ...
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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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### 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 ...
1 vote
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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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### Does there exist an almost surely a.e. differentiable continuous martingale?

Does there exist an a.s. continuous martingale $M_t$, not almost surely constant in t, that is differentiable a.e almost surely? Here the null set of non differentiability is allowed to be random, i.e....
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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 ...
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 ...
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 $I_M:=\{t_m\equiv m/M: 0\le m\le M\}$ ... 