All Questions
119 questions
12
votes
1
answer
2k
views
Hardy spaces: analysis <---> martingales
Let $H^p$ be the Hardy space of analytic functions on the open unit disk $\mathbb{D}$: $f \in H^p$ if $f$ is analytic on $\mathbb{D}$ and $\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta <...
CommunityBot
- 1
1
vote
0
answers
309
views
Horizontal vs Vertical sides Exit from a Rectangle for simple symmetric Random Walk on $\textbf{Z}^{2}$
Consider simple symmetric random walk, $X_{n} = (X_{n}^{(1)}, X_{n}^{(2)})$ with $X_0= (0,0)$, on the 2 dimensional integer lattice, $\textbf{Z}^{2}$.
Let $T_{M}, T_{N}$ be the smallest $n$ such ...
CommunityBot
- 1
10
votes
1
answer
532
views
a question on 0-1 valued stochastic process
Here's a question on probability theory from a layman (I'm a game theorist). It is very likely that the question will be a straightforward matter for someone who is a probability theorist. I guess I'm ...
- 29.8k
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
11
votes
5
answers
4k
views
Brownian motion, martingales, Markov Chains - Rosetta Stone
What are the most
fundamental/useful/interesting ways in
which the concepts of Brownian motion,
martingales and markov chains are
related?
I'm a graduate student doing a crash course in ...
- 418
4
votes
1
answer
555
views
Conditional Form of Rosenthal's Inequality
Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following:
If $\{S_i, \mathcal{F}_i, 1\leq i \leq n\}$ is a martingale and $2\...
- 3,958
4
votes
1
answer
441
views
Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?
If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
- 489
6
votes
0
answers
220
views
Reference request: Stochastic integration and martingale theory on the whole real line
I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...
- 617
1
vote
0
answers
218
views
question about Doob-Meyer decomposition
Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition:
$$V_t=V_0+\int_0^...
- 1,835
11
votes
2
answers
1k
views
Can every discrete martingale be embedded in a continuous martingale?
Let $(X_k)_{k=0,1,..., n}$ be a discrete martingale defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to know whether there exists a (continuous) martingale $(\tilde{X}...
- 724
1
vote
1
answer
411
views
a dominated convergence theorem for martingale (II)
The question is presented in
https://mathoverflow.net/questions/155392/a-dominated-convergence-theorem-for-martingale
Let $\{(X_1^n, X_2^n)\}_n$ be a sequence of martingales defined some probability ...
CommunityBot
- 1
2
votes
0
answers
519
views
asymptotic variance of sample autocorrelation of two iid random variables
I am trying to prove that the variance of the sample lag-1 autocorrelation
$$\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}$$
for an i.i.d. R.V is ...
- 171
4
votes
0
answers
274
views
Some constants in Martingale Stein inequality
Dear all,
the following is a special case of Stein inequalities for martingales.
$\textbf{Theorem}$ Let $(\Omega, \mathbb{P})$ be a (standard) probability space equipped with a filtration of ...
- 3,958
1
vote
2
answers
316
views
Martingale part of the discontinuous put payoff
I need the martingale part of the put payoff (not $C^2$..). Where $S_t=exp(\sigma W_t -\frac{\sigma^2t}{2})$
$d[(S_t -K)^+ ]$ ??
I guess I need to use local times but how?
- 24.8k
7
votes
1
answer
1k
views
a $L^1$ convergence for backward martingale
I have a question which may be naive, but I can not find the related result in the classical reference such as "Foundations of Modern Probability" and "Probability"(Billingsley). So if someone knows ...
- 3,958
3
votes
0
answers
171
views
compactness of a probability set
I have a question about the compactness of a set of martingale measures. Let $\Omega=\mathcal{C}[0,1]$ be the space of continuous functions on $[0,1]$ and $\mathcal{M}_{\Omega}$ be the family of ...
- 1,835
1
vote
0
answers
1k
views
What conditions on a filtration guarantee that a (sub)martingale has a continuous modification?
There is a theorem as follows:
Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto \...
CommunityBot
- 1
12
votes
5
answers
3k
views
Properties preserved under passage to augmented filtration
Dear all,
generally speaking, my question is about which properties of a stochastic process are preserved when I skip from the original to the augmented filtration.
Recall that if $(\mathcal{F}_t)_{...
- 1
12
votes
1
answer
5k
views
Martingales in both discrete and continuous setting
I am wondering, polynomials like
$S_n^4-6n S_n^2+3n^2+2n$ for $$S_n=\sum_{i=1}^n{X_i}$$ where $$\mathbb{P}(X_i=1)=\mathbb{P}(X_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). ...
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