All Questions
119 questions
3
votes
1
answer
476
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
3
votes
2
answers
319
views
Concentration inequality of joint event over time of a submartingale
Consider a discrete time submartingale $X_n$ with bounded difference $|X_n-X_{n-1}|\leq c$. With Azuma inequality we have the concentration of a single time event as
$$
P(X_t-X_0 \leq -t) \leq exp\...
- 185
3
votes
2
answers
229
views
Expectation of the exitpoint distance for the symmetric random walk
Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$.
Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\dotsb+X_n$, ...
- 31
3
votes
2
answers
517
views
CLT for Martingales
I posted this question originally in math stack exchange, but I got no answer.
(https://math.stackexchange.com/questions/2604591/clt-for-martingales)
In wikipedia, there is a version of a CLT for ...
- 339
3
votes
1
answer
177
views
Convergence of SDEs
Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...
- 5,405
3
votes
1
answer
181
views
When does a local supermartingale become a proper supermartingale?
This is a cross-post of my question on MSE.
Abstract: When a local supermartingale is bounded from below, is it a proper supermartingale?
Question: In remark 4.2 (p.16) of the lecture notes by Martin ...
- 235
3
votes
0
answers
147
views
Request for article in Rev. Roumaine Math. Pures Appl. (1981)
I am looking for the following article:
Al-Hussaini, A. N. A projective limit view of $L_1$-bounded martingales.
Rev. Roumaine Math. Pures Appl.26 (1981), no.1, 51–54, but I can't find it anywhere.
Do ...
- 573
3
votes
0
answers
75
views
p-Variation distance defines semi-martingales
Question
When, does the process $\tilde{X}_t$, defined path-wise by
$$
\tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right),
$$
define a ...
- 5,405
3
votes
0
answers
108
views
Has there been any study of the "extreme convergence property" for martingales?
Let $(M_n)_{n \geq 1}$ be a uniformly bounded martingale over a probability space $(\Omega,\mathcal{F},\mathbb{P})$. Define the probability measure $\mu$ on $\mathbb{R}^\mathbb{N}$ to be the law of $(...
- 708
3
votes
0
answers
124
views
How can we show that the quadratic covariation of a Hilbert space valued martingale takes values in the space of nonnegative operators?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete filtration of $\mathcal A$
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ ...
- 167
3
votes
0
answers
221
views
In which sense does the quadratic variation depend on the considered filtration?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge 0}$ be a complete right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$
$X$ be an almost surely ...
- 167
3
votes
0
answers
455
views
Hitting time of two dimensional continuous martingale
Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
- 31
3
votes
0
answers
171
views
compactness of a probability set
I have a question about the compactness of a set of martingale measures. Let $\Omega=\mathcal{C}[0,1]$ be the space of continuous functions on $[0,1]$ and $\mathcal{M}_{\Omega}$ be the family of ...
- 1,835
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 ...
- 6,213
2
votes
1
answer
301
views
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
287
views
Bernstein Inequality for continous local martingale
I'm looking for a simple proof of the following fact, which is somehow Bernstein inequality in continuous time.
Let $(M_t)_{t\geq 0}$ be a continuous local martingale. Then :
$$P\left(\sup_{t\in [0,...
- 245
2
votes
1
answer
148
views
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 ...
- 6,213
2
votes
1
answer
182
views
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
2
votes
1
answer
2k
views
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 = ...
- 6,213
2
votes
1
answer
638
views
$L^p$-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 ...
- 249
2
votes
1
answer
161
views
Concavity, martingales and stopping time
Suppose $(x_t)_t$ is a bounded $\mathbb F_t$ martingale and $f(t,x)$ is continuous, bounded, and concave in $x$. So, for any $s \ge t$, $$\mathbb E_t f(s,x_s) \le f(s,\mathbb E(x_s)) = f(s,x).$$
Does ...
- 553
2
votes
2
answers
291
views
A question about Skorokhod embedding problem
The Skorokhod Embedding Problem is well known and has many solutions. Now let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion and $\tau$ be an embedding to the centered distribution $\mu$, i.e. the ...
- 1,835
2
votes
0
answers
71
views
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}^...
- 301
2
votes
0
answers
61
views
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
2
votes
1
answer
246
views
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 ...
- 1,399
2
votes
0
answers
121
views
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
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
$$X_t:...
- 1,399
2
votes
0
answers
121
views
An unnatural martingale
What is an example of a real valued stochastic process $X$, and a filtration $\mathcal F_t$ such that $X$ is a martingale with respect to $\mathcal F_t$ but not it’s natural filtration?
Either ...
- 6,213
2
votes
0
answers
203
views
Is martingale solution equivalent to weak solution for SDE driven by stable process
Consider the following SDE
$$
d X_t=b(X_t)d t+d L_t,
$$
where $L_t$ is the symmetric $\alpha$-stable process. The corresponding generator is given by
$$
L=\Delta^{\alpha/2}+b\cdot\nabla.
$$
Is the ...
- 411
2
votes
0
answers
227
views
Non-negative martingale transforms and Radon Nikodym derivatives
Consider a filtered probability space $(\Omega, (\mathcal F_n), \mathcal F, \mathbb P)$, where $\Omega$ is the set of sequences with value in some $E \subseteq \mathbb R^d$, and $\mathcal F$ is the ...
- 341
2
votes
0
answers
110
views
Modified Pólya's Urn Process
Suppose that we have an urn that initially contains $n$ balls, partitioned into $k\geq 2$ color-classes with respect to some initial probability distribution $P=(p_1,\dots,p_k)$.
At each discrete time ...
- 396
2
votes
0
answers
440
views
Hitting time of a specific Markov chain using martingale approach (or otherwise)
Let $0 < c < 1$. Consider the Markov chain $(X_i)$ on $\{0, 1, \dots, n\}$, with transition probabilities
$$ P(k,k+1) = \left(1 - \tfrac {k}{n} \right)(1-c), \quad k = 0, \dots, n-1, $$
$$ P(k,...
- 985
2
votes
0
answers
130
views
Quadratic characteristic and constancy
Consider a change of measure on $\mathcal{F}_{t}$ defined by the restriction of two probability measures of the form
\begin{align}
\frac{dQ_{t}(\theta)}{dP_{t}}=\exp^{ \theta A_{t}-\kappa(\theta) S_{t}...
- 257
2
votes
0
answers
83
views
Modify Process to a Semimartingale
The original post is from mathstackexchange
According to some difficulties, i decided to ask here again.
Given a filtered space $(\Omega, F,\mathcal{F}_{t})$ with rightcontinous filtration. We have a ...
- 257
2
votes
0
answers
227
views
Strong law of large number for semimartingale
I just want to know if for semimartingale $X$ we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{\langle X\rangle_{t}}=0$ or when it is possible. I know it is true for Brownian motion.
Thanks
- 31
2
votes
0
answers
519
views
asymptotic variance of sample autocorrelation of two iid random variables
I am trying to prove that the variance of the sample lag-1 autocorrelation
$$\hat{\rho}=\frac{\sum_{t=1}^n(x_t-\bar{x})(x_{t-1}-\bar{x})}{\sum_{t=1}^n(x_{t-1}-\bar{x})^2}$$
for an i.i.d. R.V is ...
- 171
2
votes
0
answers
134
views
Supermartingale inequality on a particular event
Say, I have a supermartingale $Y_t$ with respect to the filtration $F_t$. Let $T$ and $S$ two stopping times greater than $t>0$ such that on the event $A$, $T>S$, then since $Y_t$ is a ...
- 23
1
vote
1
answer
185
views
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
1
vote
1
answer
182
views
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)...
- 335
1
vote
2
answers
316
views
Martingale part of the discontinuous put payoff
I need the martingale part of the put payoff (not $C^2$..). Where $S_t=exp(\sigma W_t -\frac{\sigma^2t}{2})$
$d[(S_t -K)^+ ]$ ??
I guess I need to use local times but how?
- 111
1
vote
2
answers
3k
views
Is stopped brownian motion not a martingale?
In page 45 of the book "Financial Derivatives In Theory and Practice by P.J.Hunt and J.E.Kennedy, it seems to me that the author says the stopped Brownian Motion is not a martingale as follows.
(...
- 19
1
vote
1
answer
1k
views
Predictable quadratic Variation <.> has same intervals of constancy as the process
From
Revuz and Yor - Continuous Martingales and Brownian Motion 1999
Chapter IV Proposition 1.13
it is proven, that for a continuous local martingale $M_t$ the intervals of constancy ...
- 257
1
vote
1
answer
411
views
a dominated convergence theorem for martingale (II)
The question is presented in
https://mathoverflow.net/questions/155392/a-dominated-convergence-theorem-for-martingale
Let $\{(X_1^n, X_2^n)\}_n$ be a sequence of martingales defined some probability ...
- 1,835
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(\...
- 6,213
1
vote
1
answer
139
views
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
1
vote
1
answer
419
views
Proof of the existence of the covariation of a continuous local martingale presented in the book of Kallenberg
Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\in[0,\:T]}$ be a complete filtration on $(\Omega,\mathcal A)$
$M$ be an almost surely continuous local $\...
- 167
1
vote
1
answer
687
views
Supremum in a Markov chain model
A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...
- 21
1
vote
0
answers
125
views
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 ...
- 1,399
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
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