All Questions
119 questions
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). ...
- 255
12
votes
3
answers
2k
views
Compactness of the set of densities of equivalent martingale measures
Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the set $\mathcal P^\...
- 243
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)_{...
- 121
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 <...
- 943
12
votes
1
answer
330
views
Convergence of an implicitly defined sequence of random variables
Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
- 171
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}...
- 1,835
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 ...
10
votes
2
answers
828
views
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
10
votes
1
answer
2k
views
Law of large numbers for martingales
I apologize in advance if this question is too basic, but I've received no response on Math Stack Exchange, so perhaps it is more appropriate here:
Let $X_n$ be a square-integrable martingale with $\...
- 103
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 ...
10
votes
4
answers
680
views
The min of the mean of iid exponential variables
Let $X_1, \ldots, X_n, \ldots$ be iid exponential random variables with mean 1. It is well-known that $\min_{1\le j < \infty} \frac{X_1 + \cdots + X_j}{j}$ follows the uniform distribution U(0,1). ...
- 773
9
votes
3
answers
448
views
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
9
votes
1
answer
556
views
Berry-Esseen bound for martingale sequence with varying and dependent variances
Let $(X_{1},\ldots,X_{k},\ldots)$ be a martingale difference sequence, i.e.
$$
E[X_{k}|\mathcal{F}_{k-1}] = 0
$$
where $\mathcal{F}_{k-1}$ is the $\sigma$-algebra filtration at $k-1$.
Let $\sigma_{...
- 719
9
votes
1
answer
4k
views
Quadratic variation and predictable quadratic variation for martingales
Let $(M_{t})_{0\le t\le 1}$ be a continuous martingale with respect to the filtration $(\mathcal{F}_{t})_{0\le t\le 1}$. Assume that $E M_1^2<\infty$.
Fix $N$ and consider now a discrete version ...
- 931
8
votes
3
answers
2k
views
What is the optimal growth of the constant in BDG?
Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} |X|^...
- 4,010
8
votes
1
answer
694
views
A generalization of Jensen's Inequality
Jensen's inequality is well known as
$$E\big[f(X)\big]\le f\big(E[X]\big)$$
where $X$ is a integrable random variable and $f: R\to R$ is a bounded concave function, see also http://en.wikipedia.org/...
- 1,835
7
votes
2
answers
2k
views
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 ...
- 6,215
7
votes
2
answers
594
views
Large deviation/concentration inequality for submartingale
Let $S_t = M_t + D_t$ be the sum of a martingale $\left(M_t\right)_{t=1,2,\ldots}$ and a predictable process $(D_t)_{t=1,2,\ldots}$ such that the variance of the increments of $M$ is uniformly bounded ...
- 355
7
votes
2
answers
2k
views
Proof of extended supermartingale convergence theorem
There is a supermartingale convergence theorem which is often cited in texts which use Stochastic Approximation Theory and Reinforcement Learning, in particular the famous book "Neuro-dynamic ...
- 183
7
votes
1
answer
1k
views
Moment bounds on exponential martingale
Consider the exponential martingale used in the Girsanov transformation of
measure:
$$Z(t) = \exp\Big(\int_0^tXdW - \frac{1}{2}\int_0^t|X|^2ds\Big)$$
so that $Z$ solves the sde $dZ = ZXdW$ where $W$ ...
- 143
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 ...
- 1,835
7
votes
1
answer
409
views
Do i.i.d. sums concentrate any faster than martingales?
Suppose $X_1,X_2, \ldots, X_N \in \mathbb R^d$ are random variables with each $\|X_n\|_2 \le 1/2$ (this choice of the constant simplifies later formulae).
The simplest concentration inequality I know ...
- 1,955
7
votes
1
answer
487
views
A note on Doob's theorem
I have faced the following problem, regarding to the Martingale Theory. Because this area far from my area I don't know whether this problem is in literature or this can be simple question for ...
- 103
7
votes
1
answer
560
views
Doob's inequality for martingale "convolution"
Let $(X_t, t \in \mathbb{N})$ be a martingale, and let $a \leq b \leq T \in \mathbb{N}$ be constants. Is there something like Doob's inequality for $\mathbb{E} \sup_{a \leq t \leq b} X_t(X_T-X_t)$, i....
- 1,069
6
votes
3
answers
999
views
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?
- 6,215
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 ...
- 6,215
6
votes
3
answers
2k
views
Iterated Ito Integral, Gaussian Volterra Process
Let me define
$$
J^f_{n}(t) = \, \int_0^t \int_0^{t_1} \ldots \int_0^{t_{n-1}} f(t, t_1, \ldots, t_n) \; dB_{t_n} ...dB_{t_1}
$$
where $f:[0,1]^{n+1} \to \mathbb{R}$ is a nice deterministic function....
- 271
6
votes
1
answer
660
views
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 ...
- 6,215
6
votes
1
answer
461
views
Gronwall lemma with conditional expectation
The discrete Gronwall's inequality states that if $x_n$ and and $u_n$ are non-negative sequences such that
$$ x_{n+1}\le a+\sum_{k=0}^n u_k x_k$$
then $$x_n\le a\prod_{k=0}^{n-1} (1+u_k)$$
(It can be ...
- 562
6
votes
0
answers
150
views
Delayed Pólya's urn process
The standard Pólya's urn process can be stated as follows:
You have an urn with red and green balls. At any time unit you choose one ball at random, note the colour, and give the ball back. At the ...
- 396
6
votes
0
answers
183
views
Distribution of the stopping time of an autoregressive sequence
Consider $e_t$ being i.i.d. uniformly chosen from $\pm 1$. Let $\eta$ be a small positive constant. What is the distribution of $T$ such that $\eta^{0.5} (1+\eta)^T W_T$ first hits $\pm 1$, in which
$$...
- 1,127
6
votes
0
answers
220
views
Reference request: Stochastic integration and martingale theory on the whole real line
I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...
- 617
5
votes
4
answers
1k
views
Examples of discrete time martingales
In probability, a martingale is given by a sequence of integrable
random variables $(S_n)$ and an increasing sequence of
$\sigma$-algebras ${\cal F}_n$ such that
$S_n$ is ${\cal F}_n$-...
- 18.7k
5
votes
1
answer
652
views
Proof of Pinelis (1992) - Banach space inequalities
I am reading Pinelis "An approach to inequalities for the distributions of infinite -dimensional martingales" and cannot follow his proof of Theorem 3:
Let $(f_n)$ be a martingale in a separable ...
- 619
5
votes
2
answers
311
views
A comparison of diffusions
Consider two diffusions given by
$$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$
for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough ...
- 128k
5
votes
1
answer
208
views
Expected supremum of normalised random walk
Let $X^i\in \mathbb R^d$ be iid. random variables for $i=1$ to $n$.
Assume $\mathbb E[X^i]=0$ and the covariance matrix $\mathbb C[X^i] = \mathbb E[X^iX^{iT}] = I$ is the identity matrix.
Define $S^k=...
5
votes
1
answer
350
views
Can an a.s. non constant continuous martingale be differentiable with nonzero probability?
Let $M$ be a continuous martingale such that almost surely, the sample paths of $M$ are not constant.
Question: Is it true that $M$ is almost surely not differentiable?
- 6,215
5
votes
1
answer
1k
views
Supremum of a martingale
Let $(X_n)$ be a martingale. What can be said about the distribution of its maximum over a window of fixed length:
$$M_n = \max_{n-10 \leq k \leq n} X_k$$ or about the "range" over a window:
$$R_n = \...
- 587
5
votes
0
answers
653
views
Explicit martingale representation for a Brownian bridge
Let $W$ denote a Wiener process, $\displaystyle M_t = \max_{0 \le s \le t} W_s$ its running maximum. The martingale representation of $M$ is known explicitly:
$$M_T = \sqrt{\frac{2T} \pi} + \int_0^T ...
- 341
4
votes
2
answers
373
views
Another curious martingale
This is a natural follow up question to A curious martingale.
Does there exist an almost surely continuous martingale that converges in probability to $+\infty$?
Note: We say a process $X_t$ converges ...
- 6,215
4
votes
1
answer
677
views
If the moving average of a process is a martingale, is the process a martingale?
Problem set up:
Let $\mathcal F_t$ be a filtration satisfying the usual conditions. Let $T > 0$ be a fixed real number, and define the filtration $\mathcal H_t := \mathcal F_{T + t}$.
Suppose a ...
- 6,215
4
votes
1
answer
555
views
Conditional Form of Rosenthal's Inequality
Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following:
If $\{S_i, \mathcal{F}_i, 1\leq i \leq n\}$ is a martingale and $2\...
- 289
4
votes
1
answer
441
views
Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?
If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
- 235
4
votes
1
answer
262
views
Bounded density for diffusions with diffusion coefficients bounded away from $0$
Consider a diffusion given by
$$X_t=\int_0^t a(s,X_s)\,dW_s$$
for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
- 128k
4
votes
1
answer
594
views
Martingales and intersection of random walks
Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...
- 519
4
votes
0
answers
80
views
Does this filtration have a name?
In the context of Ethier&Kurtz Markov Processes: Characterization and Convergence (Chapter 4, equation (3.2)) as well as the two papers Martingale problems for conditional distributions of Markov ...
- 183
4
votes
0
answers
274
views
Some constants in Martingale Stein inequality
Dear all,
the following is a special case of Stein inequalities for martingales.
$\textbf{Theorem}$ Let $(\Omega, \mathbb{P})$ be a (standard) probability space equipped with a filtration of ...
- 769
3
votes
1
answer
543
views
Stochastic integrals as honest martingales -- comparison criterion
We have a given positive martingale $\rho_t$, with the dynamics:
$$\textrm{d} \rho_t = \lambda_t \rho_t \textrm{d} W_t$$
where $W_t$ is a standard Brownian motion. Now we have a "dumped" process p_t:
$...
- 667
3
votes
1
answer
181
views
A nice terminal inequality for martingales
Let $X_t$ be a continuous time martingale taking with $\sup_t \mathbb E[X_t^-] < \infty$, and $X_0 = 0$ almost surely. Assume further that $X_1$ admits a probability density function.
Is it true ...
- 6,215
3
votes
1
answer
824
views
Stochastic integrals as honest martingales — exponential damping
We have a given positive martingale ρt, with the dynamics:
$$\textrm{d}\rho_t = \lambda_t \rho_t \textrm{d}W_t$$
where $W_t$ is a standard Brownian motion. Now we have an "exponentially dampened" ...
- 667