# Questions tagged [martingales]

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241
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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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111
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### Stopping times for martingale

The nonnegative integer set is denoted by $\mathbb{Z}_+$.
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space and $\{\mathcal{F}_{n}\}_{n\in{\mathbb{Z}_+}}$ be an increasing sequence ...

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 ...

0
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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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141
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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 \...

2
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49
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### Orthogonal basis of martingale-Hardy space

Let $(X,\mathcal{B}(X),(\mathcal{A}_n)_{n=1}^N,\mu)$ be a filtered probability space with $\mathcal{A}_N=\mathcal{B}(X)$ and let $H$ be the space of $\mathcal{A}_{\cdot}$-adapted martingales $m_{\cdot}...

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1
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51
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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
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1
answer

144
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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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174
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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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169
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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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48
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### Inequality for moment of Hawkes process

During research on the Hawkes process, I face the following problem.
We suppose to have a Hawkes process $N_t$ whose intensity follows
$$
\lambda_t = \mu + \underbrace{\phi* dN_t}_{\int_0^t\phi(t-s)...

1
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1
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162
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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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271
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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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132
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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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### Reference request : Is there "equilibrium point" for (martingale) stochastic differential equations?

It is known that ODEs have the so-called equilibrium point, i.e. $x(t)\equiv x^*\in\mathbb R$ is an equilibrium point of $x'(t)=f(t,x(t))$ if $f(t,x^*)\equiv 0$. Consequently, an study of the ...

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41
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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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88
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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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131
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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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1
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80
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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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78
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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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286
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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 ...

3
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2
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188
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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 $...

5
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2
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215
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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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124
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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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0
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201
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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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101
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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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192
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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 $\mathcal H_\infty$-...

3
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0
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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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173
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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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0
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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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165
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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 $I_M:=\{t_m\equiv m/M: 0\le m\le M\}$ ...

2
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1
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### $L^p$-convergence of submartingale

Let $p\geq1.$ Consider a $\mathcal{F}_k$-submartingale $(X_k)_k$ in $L^p.$ We can prove easily that $(X_k)_k$ converges in $L^p$ if and only if $(|X_k|^p)_k$ is uniformly integrable.
If $(X_k)_k$ was ...

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### Does there exist an almost surely differentiable martingale?

Does there exist a continuous time martingale $X_t$ not a.s. constant in $t$ that is almost surely everywhere differentiable?

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### Identification of some continuous Markov martingale

For integers $M,N\ge 1$, set $I_M:=\{t_m\equiv m/M: 0\le m\le M\}$ and $J_N:=\{x_n\equiv n/N: -N\le n\le N\}$. We define a discrete Markov martingale $(X^{M,N}_t)_{t\in I_M}$ taking values in $J_N$ ...

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### A semimartingale interpolation problem

This question is a direct extension of this one.
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a stochastic basis and let $N\in\mathbb{Z}^+$, $T>0$, $\{t_n\}_{n=1}^{N}$ be a ...

1
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1
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### Does a sequence that verifies the assumptions of a square integrable martingale on some event need to be convergent on this event?

I came across this claim by reading some literature on stochastic approximation.
Let $(\Omega, \mathcal{A}, \mathbb{P}$) be a probability space, $(\mathcal{F}_n)$ a filtration on it. Let $(\epsilon_{n}...

2
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1
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133
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### If a process is periodic on average with mutually incommensurable periods, is the process a martingale?

Motivation:
If a continuous function on the real line is periodic with periods $p_1, p_2 > 0$ such that $\frac{p_1}{p_2}$ is irrational, then the function is constant. Is there a probabilistic ...

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286
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### If the moving average of a process is a martingale, is the process a martingale?

Problem set up:
Let $\mathcal F_t$ be a filtration satisfying the usual conditions. Let $T > 0$ be a fixed real number, and define the filtration $\mathcal H_t := \mathcal F_{T + t}$.
Suppose a ...

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1
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92
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### Weaker than martingale condition

Let $\mathcal{F}_n$ be a filtration and $S_n$ be a sequence such that $\mathbb{E}[S_n-S_{n-1}|\mathcal{F}_{n-2}]=0$ for all $n$. This condition is similar to the martingale condition but the ...

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### Decomposition of reversed processes

Consider a reversed filtration $(\mathcal{F}_k)_{k \geq 0} $ $(\mathcal{F}_{k+1} \subset\mathcal{F}_k),$ $(X_k)_{k \geq0}$ is a processes in $L^1,\mathcal{F}_k$-adapted.
Is it possible to decompose $...

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### On the property of a nonnegative stochastic process "attracted" near zero

Let $\{X_k\}$ be a nonnegative stochastic process satisfying
$$E\left[ X_{k+1} \mid \mathcal{F}_k \right] \leq \rho X_k + c,$$
where $0 < \rho < 1, c>0$. Intuitively, the process is likely to ...

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### Almost supermartingale and a.s convergence

After reading a paper on the convergence of almost supermartingale, the following result appeared:
If $(X_k)_k,(Y_k)_k,(W_k)_k$ are three $(\mathcal{F}_k)$-adapted processes taking values in $\mathbb{...

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84
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### Moment generating function of a stopped process from Wald's identity

In an exercise I am asked to prove the following Wald's identities: let $S_n$ be a simple random walk and $T$ a stopping time. Then for all $\lambda \in \mathbb R,$
$$
\mathbb E(e^{\lambda S_1}) = 1 \...

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### How can we use Martingales to identify an unknown particle?

Suppose there is a particle in a box. We are interested in identifying what type of particle it is, but are not allowed look inside the box. All we can do is observe the particles that are entering ...

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### Martingales associated with heat equation

I am trying to learn the connection between Brownian motion and heat equation (in the spirit of Feynman-Kac, for example, here). I read (Michael E. Taylor's PDE book, Volume II, Chapter 11, ...

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### Is a stopped Ito-integral integrable if the Ito integrand is only square-integrable on an open interval?

Assume a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\in[0;T)}, \mathbb P)$ with an $\mathbb R^n$-valued Brownian motion $\{W_t\}_{t\in[0;T)}$ and the filtration $\{\mathcal F_t\}_{t\in[0;T)...