# Questions tagged [martingales]

The martingales tag has no usage guidance.

255
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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 ...

2
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1
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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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0
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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\{...

2
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0
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243
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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:...

0
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1
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124
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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 $...

1
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1
answer

105
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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 ...

0
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0
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140
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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
...

3
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1
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105
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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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467
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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 ...

3
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0
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56
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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 $\...

1
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1
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222
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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 ...

2
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1
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224
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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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136
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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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0
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81
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### Martingale w.r.t. filtration

Let $(\Omega,\mathcal{F},\mathbb{P})$ be some probability space and $(\mathcal{F}_{t})_{t \in \mathbb{N}}$ be a filtration over $(\Omega,\mathcal{F})$. Let $X : \Omega \mapsto \mathbb{R}$ be any $\...

2
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1
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121
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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 ...

2
votes

1
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131
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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
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1
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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 ...

2
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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 ...

2
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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 ...

2
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1
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506
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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 = ...

3
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1
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232
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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 \...

0
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1
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59
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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 ...

2
votes

1
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219
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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 ...

1
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1
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181
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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 ...

5
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1
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298
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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?

1
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1
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189
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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 ...

10
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1
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379
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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....

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0
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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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### 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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1
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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 ...

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1
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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 $\operatorname E[\varphi_i\mid\mathcal A_{-n}]\to\operatorname E[\varphi_i\mid\mathcal A_{-\infty}]$ ...

2
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### 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]=\...

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0
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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 ...

2
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1
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435
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### 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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2
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### 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 $...

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2
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### 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 ...

1
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1
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### 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
$$X_t=1+\int_0^t a(s,X_s)dW_s,\quad \forall t\ge 0.$$
...

2
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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{...

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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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1
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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 ...

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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 ...

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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 ...

0
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1
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182
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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 $I_M:=\{t_m\equiv m/M: 0\le m\le M\}$ ...