All Questions
Tagged with pr.probability martingales
210 questions
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Characterisation of a family of continuous martingales
I look for a full characterisation of the continuous martingales $X=(X_t)_{0\leq t\leq T}$ (defined on some filtered probability space as nice as possible) such that
$$X_0=0\quad \mbox{ and } \quad\...
- 1,389
0
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0
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31
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Looking for a citation for this simple generalization of the Markov bound to non-negative super-martingales
Does anybody know a reference for the following theorem?
Theorem 1. Let $(X_t)_{t=0}^\infty$ be a non-negative supermartingale.
Then, for any constant $c > 0$, the event $(\exists
> t)\, X_t \...
- 183
4
votes
1
answer
66
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Expectation bounds on supremum of family of martingales
Suppose I fix a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, P)$ and on it a Brownian motion $B$. Let $\tau_\alpha$ denote a set of stopping times which satisfies $\sup_\alpha \tau_\...
- 143
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1
answer
57
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Lower bounding an alternating series with signs from a martingale difference sequence
Let $\epsilon_n \in \{-1, 1\}$ be a martingale difference sequence, in the sense that
$$M_n := \sum_{i = 0}^n \epsilon_i$$
is a martingale.
We assume $\epsilon_0 = \pm 1$ with probability $\frac{1}{2}$...
- 6,165
6
votes
1
answer
659
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On the martingale betting scheme
For a fixed probability $0 < p < 1$, let $X^p$ be the martingale that goes up by $1$ with probability $p$, and goes down by $\frac{p}{q}$ with probability $q := 1-p$.
Write $X$ for the ...
- 6,165
1
vote
1
answer
60
views
Reverse Doob’s maximal inequality for bounded martingales
Consider the set of discrete or continuous time $L^\infty$-bounded martingales $X$ with $X_0 = 0$ almost surely. Here $L^\infty$-bounded means $\|X\|_{\infty} := \sup_t \mathbb \|X_t\|_{L^\infty(\...
- 6,165
3
votes
1
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181
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A nice terminal inequality for martingales
Let $X_t$ be a continuous time martingale taking with $\sup_t \mathbb E[X_t^-] < \infty$, and $X_0 = 0$ almost surely. Assume further that $X_1$ admits a probability density function.
Is it true ...
- 6,165
0
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1
answer
64
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Sharpening Doob’s upcrossing inequality for Brownian motion
Note: This question is heavily related to a series of posts ([1], [2]) by user GJC20.
Provided a martingale $X$ in continuous-time, Doob's upcrosssing inequality states:
If $U(a,b)$ denotes the number ...
- 6,165
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2
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60
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Do continuous martingales satisfy this nice terminal inequality?
Let $X$ be a continuous, non negative martingale on $[0, 1]$ with $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. Assume further that $X_1$ admits a probability density function. Is it true that the ...
- 6,165
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0
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71
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Assumptions Wald's second equation?
Let $(X_n)_{n\in \mathbb{N}}$ be an i.i.d. sequence of random variables and $N$ an $\mathbb{N}_0$ valued random variable. Let $X_1 \in \mathcal{L}^2$ and $N \in \mathcal{L}^1$. Let $S_n := \sum_{i=1}^...
- 261
1
vote
1
answer
185
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Sum of $X_k$ with $\mathbb{P}(X_k=\pm 1) = 1/2\pm 1/(2\sqrt{k})$
Let $\{X_k\}$ be a sequence of mutually independent random variables with
\begin{align}
\mathbb{P}(X_k = 1) & = \frac{1}{2} + \frac{1}{2\sqrt{k}},
\\
\mathbb{P}(X_k = -1) & = \frac{1}{2} - \...
- 269
3
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0
answers
101
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Divergent/Unbounded random walks techniques
I want to prove the following biased random walk will be diverge. Suppose I have a random walk $S_n = X_1 + ... + X_n$, but $X_1,...,X_n$ are dependent variables. $X_1 \sim$ Bernoulli($\sigma(\theta_1)...
- 31
9
votes
3
answers
448
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All stationary martingales are constant?
Suppose $(X_{n})_{n\geq{1}}$ is a stationary process that is a martingale with respect to some filtration. Suppose also that $\mathbb{E}X_{0}^{2}<\infty$ so that $\mathbb{E}X_{n}^{2}<\infty$ for ...
- 662
2
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0
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61
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Characterisation of Bessel process
Let $\delta \in (0, 2)$; $(X_t)_{t \ge 0}$ a nonnegative continuous Markov process. Suppose that
For each $T \ge 0$, if we write $\tau \overset{\mathrm{def}}= \inf\{t \ge T : X_t = 0\}$, then $(X_{T +...
- 177
3
votes
0
answers
80
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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 ...
- 353
1
vote
1
answer
129
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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, ...
- 353
3
votes
1
answer
180
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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 ...
- 235
3
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1
answer
553
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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 $\...
- 61
4
votes
0
answers
80
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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 ...
- 183
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0
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90
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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 ...
- 573
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0
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125
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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 ...
- 1,389
6
votes
1
answer
396
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 ...
- 6,165
2
votes
1
answer
246
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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 ...
- 1,389
10
votes
2
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828
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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 ...
- 128k
4
votes
2
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373
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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 ...
- 6,165
7
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2
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2k
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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 ...
- 6,165
2
votes
1
answer
74
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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$ ...
user514580
3
votes
0
answers
147
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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 ...
- 573
2
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0
answers
121
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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 ...
- 573
1
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1
answer
83
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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}...
- 113
1
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0
answers
156
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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
$$
T_t=\inf \left\{...
- 121
2
votes
0
answers
282
views
Identify two continuous martingales in law as time-changed Brownian motions
Let $W$ be a Brownian motion and $\alpha$ be a progressively measurable process taking values in $\mathbb R_+$. Set $\beta_t:=\max(\alpha_t, 1)$ for all $t\ge 0$. Define respectively $X$, $Y$ by
$$X_t:...
- 1,389
1
vote
1
answer
464
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 $...
- 95
3
votes
1
answer
352
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 $\tau_a$ = $\...
- 33
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0
answers
87
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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 ...
- 2,207
8
votes
1
answer
533
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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 ...
- 83
6
votes
1
answer
168
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 $x\in[0,1]$...
- 715
1
vote
0
answers
182
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 $\...
- 573
2
votes
1
answer
198
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 ...
- 6,165
2
votes
1
answer
182
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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 ...
- 335
1
vote
1
answer
139
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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 ...
- 573
2
votes
1
answer
2k
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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 = ...
- 6,165
3
votes
1
answer
474
views
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 \...
- 53
5
votes
1
answer
350
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?
- 6,165
10
votes
1
answer
700
views
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....
- 34.7k
1
vote
0
answers
240
views
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 ...
- 11
2
votes
1
answer
300
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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 $\operatorname E[\varphi_i\mid\mathcal A_{-n}]\to\operatorname E[\varphi_i\mid\mathcal A_{-\infty}]$ ...
- 167
2
votes
1
answer
274
views
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
$$ \mathbb{E}\big[Y_n^2-Y_{n-1}^2\big]=\...
- 23
4
votes
1
answer
262
views
Bounded density for diffusions with diffusion coefficients bounded away from $0$
Consider a diffusion given by
$$X_t=\int_0^t a(s,X_s)\,dW_s$$
for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
- 128k
5
votes
2
answers
311
views
A comparison of diffusions
Consider two diffusions given by
$$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 ...
- 128k