# Tagged Questions

A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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116 views

### Does a theory of stochastic differential algebras exist?

My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...

**13**

votes

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355 views

### Why, and how badly, does the proof of “no percolation at the critical point in half-spaces” fail for full spaces?

The proof by Barsky et. al. that there is no percolation in half-spaces proceeds by a dynamic renormalization argument. The proof couples critical percolation in the half-space $\mathbb{H}^d$ with a ...

**7**

votes

**0**answers

57 views

### Distribution of the stopping time of an autoregressive sequence

Consider $e_t$ be 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
$$
...

**7**

votes

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545 views

### Do isonormal Gaussian processes have measurable sample paths?

Let $H$ be a real separable Hilbert space. Let $W=\{W(h):h\in H\}$ be a real-valued stochastic process defined on a complete probability space $(\Omega,\mathcal{F},P)$. Assume that $W$ is a centered ...

**6**

votes

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148 views

+50

### Hierarchical (Recursive) Random Walk Model

Consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/...

**6**

votes

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259 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....

**6**

votes

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437 views

### When is an ODE a good approximation to an SDE?

Suppose $X_t$ is a weak solution to a stochastic differential equation in the form
$$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$
for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb R^...

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votes

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223 views

### Does the law of a Feller Process on a non-locally-compact Polish space depend continuously on the initial condition (in Skorohod path-space)?

I am sure this is written down somewhere but cannot find it.
Consider a Polish space $E$ and a strong Markov process $(X_t)_{t\ge 0}$ with values in $E$ and cadlag paths. More precisely, we have a ...

**6**

votes

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385 views

### Do there exist generalized conformal maps that preserve elliptic measure?

Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction $f:...

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443 views

### Is there a continuous-time version of Kingman's subadditive decomposition theorem?

Kingman's subadditive ergodic theorem (see this article) states that if $x_{m,n}$ is a real valued process indexed on the set of pairs of non-negative integers $m < n$ satisfying:
$x_{l,n} \le x_{...

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votes

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192 views

### Exponential tails for a functional of a subcritical branching process.

Let $(m_i, i \in \mathbb{N})$ be positive weights with $\sum_{i \in \mathbb{N}} m_i^2 < 0.1$.
Consider a subcritical branching process in discrete time and continuous space,
started from some ...

**5**

votes

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82 views

### Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term

Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...

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votes

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69 views

### Most visited vertex in a random walk with place dependent drift

Consider the following Markov chain on $\mathbb{Z}$:
$$
P(x,x+1)=1-P(x,x-1)=\frac{1}{2}+e^{-|x|}\cdot \mathbf{1}_{\{x\neq 0\}}
$$
Do there exist constants $c,C>0$ such that
$$
c\cdot P^t(z,z) \...

**5**

votes

**0**answers

49 views

### Full distribution of FPTs in random walks on graphs

There is a lot of published research on the mean passage passage time (FPT) for random walks on various types of graphs. How about the variance of the FPT and higher momenta? In fact, I would be ...

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votes

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432 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 ...

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votes

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270 views

### Squaring random Schwartz distributions

Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance
$$
\mathbb{E}
[\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)}
\widehat{g}(\xi)}{|\xi|^{d-2[\...

**5**

votes

**0**answers

115 views

### Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?

Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mu$ be a probability measure ...

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votes

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190 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 ...

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votes

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124 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 ...

**5**

votes

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529 views

### Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...

**5**

votes

**0**answers

351 views

### Skorohod theorem (weak convergence) on a discrete setting

I have a question about the application of Skorohod representation theorem. The questions arises in this paper about robust hedging in mathematical finance. It is about the very last equation on page ...

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votes

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147 views

### Local structure in the stochastic sandpile model

Here's a question that came up at the recent AIM conference on chip-firing and generalizations.
The stochastic sandpile model, I think originally due to Manna, is a stochastic process that (in one ...

**5**

votes

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181 views

### Constructing black noise with non-standard analysis

With noise in the sense of i.i.d. random sequence,
a noise is black if it is not isomorphic to standard Gaussian white noise.
Tsirelson showed the existence of black noise through the scaling limit ...

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votes

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123 views

### Implications of Half-Space Percolation

Let $\mathbb{Z}^d$ be the usual $d$-dimensional lattice and let $\mathbb{H}:=\mathbb{Z}^{d-1}\times Z_+$, where $Z_+:=[0,1,2,\ldots]$. If we now consider bond percolation on $\mathbb{H}$, it is a ...

**5**

votes

**0**answers

1k views

### Levy jump measure vs. Levy measure vs. sum of jumps

This question might be a bit basic, but I am struggling to understand the connection between various versions of the Ito's lemma for Levy processes (and semimartingales in general). Could someone ...

**4**

votes

**0**answers

188 views

### Stopping times for Brownian motion

Let $B_t, t\geq 0$ be standard Brownian motion.
Let $\big(\mathcal{G}_t, t\geq 0\big)$ be the natural filtration, defined by $\mathcal{G}_t=\sigma(B_s, 0\leq s\leq t)$.
Define also a filtration $\...

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votes

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67 views

### I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...

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votes

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157 views

### Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories

Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...

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votes

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194 views

### Self-adjusting random walk

Let $X_t$ be a random process such that
\begin{eqnarray}
X_1 &=& 0\\
X_t &=& X_{t-1} + \left\{\begin{array}{ll}
A_t, & X_{t-1} \geq 0\\
B_t, & X_{t-1} < 0
\end{array}\...

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votes

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168 views

### Existence or construction of a sequence of orthogonal matrices with three properties

This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help ....
Any pointers or suggestions are appreicated!
...

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votes

**0**answers

107 views

### regularity of zero point

We consider 1-d process $X$
$$ X(t) = b t + J_{t} + M_{t}$$
where $b$ is constant, $M$ is a continuous martingale process with
$M(0) = 0$, and
$J$ is a symmestric $\alpha$-stable process with its ...

**4**

votes

**0**answers

50 views

### Existence of martingales given some constraint on laws

Let $X=(X)_{0\le t\le 1}$ be a continuous martingale starting at $0$, then denote by $\mu$ and $\nu$ the probability laws of $\int_0^1X_t \mathrm{d}t$ and $X_1$. Then it is easy to see that the couple ...

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votes

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87 views

### Is there a generalization of Polya urns to continuous outcome event?

Take for example the simplest model where there are n blue balls and m white balls in an urn. Then, in a first step realization, a white one has been drawn and then c + 1 of this colour had been put ...

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votes

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61 views

### Cycle removal process

Consider the following stochastic process for generating a forest: start from a complete graph on $n$ vertices and proceed to repeatedly remove the edges of uniformly chosen cycles. Formally, let $G_0$...

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votes

**0**answers

102 views

### Using Crump-Mode-Jagers processes to get logarithmic bound on a random tree height

I am currently pursuing my PhD degree and in my research I came across a family of random trees. I need to prove a logarithmic asymptotic bound for the heights of such trees as their size grows. I ...

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votes

**0**answers

211 views

### Generalized Markov Processes on CW complexes of dimension > 1

Markov processes have a large variety of applications to physics and chemistry (as well as many other fields). Such processes are formulated on graphs, i.e., CW complexes of dimension one. It is ...

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votes

**0**answers

73 views

### Sufficient condition in terms of stopping times for a stochastic process to be a local supermartingale

Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin (a,b)\}$. We can interpret $\tau_{s,a,b}$ as the first ...

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votes

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485 views

### Inverse Fourier Transform involving a Bessel Function, Exponential, and Power

I'm interested in this integral as a function of $r$ for various spectral densities $S(s)$:
$\frac{2 \pi}{r^{p/2}-1} \int_{0}^{\infty} S(s) J_{p/2-1}(2 \pi r s) s^{p/2} ds $, where $J_{p/2-1}$ is a ...

**4**

votes

**0**answers

73 views

### Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...

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votes

**0**answers

95 views

### How fast is discrete-time diffusion on a continuous set?

This question is inspired by Joseph O'Rourke's beautiful answer to my previous question.
Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns have unit norm and sum ...

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votes

**0**answers

230 views

### Inadmissibility of Simpson's rule

(An earlier version of this at stackexchange got no answers.)
Bayesianism says that all uncertainties, or at least all uncertainties about the truth or falsity of propositions, can be expressed by ...

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votes

**0**answers

86 views

### Sufficiency of stationary policy for negative stochastic dynamic programming

Consider a Markov Decision Process with Borel state space $X$ and Borel action space $U$, like the one defined in the book "Stochastic Optimal Control: Discrete-time case" by Bertsekas and Shreve. All ...

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votes

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306 views

### Integrating a Bessel Bridge

Preliminaries
An order-3 Bessel Process is the one-dimensional stochastic process $X$ described by $X(t) = \sqrt{W_1(t)^2 + W_2(t)^2 + W_3(t)^2}$, where each $W_k$ is an independent Brownian Motion. ...

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votes

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230 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 ...

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votes

**0**answers

658 views

### The spectrum of a Markov Operator and Invariant Measures

Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...

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votes

**0**answers

201 views

### stochastic control / geometric mean

Consider the following problem:
Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...

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votes

**0**answers

60 views

### “Local” functional central limit theorem for the empirical distribution function

This question is a repost from Mathematics Stack Exchange, where it did not receive any answer.
Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. real-valued random variables such that $\mathbb E[...

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votes

**0**answers

43 views

### Feynman-Kac formula and time-ordering for vector bundles

Let $M$ be a compact Riemannian manifold and let $\mathrm{d}\mathbb{W}^{yx;T}(\gamma)$ denote the Brownian Bridge measure, i.e. the Wiener measure on the paths that travel from $x$ to $y$ in time $T$ (...

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votes

**0**answers

42 views

### A question on improper Itô integrals and semimartingales

I am reading the article given in http://link.springer.com/chapter/10.1007/978-1-4614-5906-4_24#page-1. I have the following two questions:
In which setting does one define improper integrals with ...

**3**

votes

**0**answers

39 views

### Memorylessness of residence times for a Markov process

I'm stuck on the trivial problem of showing memorylessness of holding (residence) times for a continuous time homogeneous Markov chain on finite state space.
I have a homogeneous Markov process $x(t),...