# Tagged Questions

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

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### Does the martingale property holds after changing filtration?

Let $\Omega$ be the space of continuous real-valued functions $\omega=(\omega_t)_{t\ge 0}$ starting at zero, i.e. $\omega_0=0$. Let $\Lambda=\Omega\times \mathbb R_+$ and denote by $\lambda=(\omega,\...

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26 views

### Characterisation of non-Gaussian stationary stochastic processes via auto-correlation functions

It is well-known that a centred stationary Gaussian stochastic process is characterised up to equivalence by its autocorrelation function.
Wiener, in his Time Series, makes the off-hand remark that ...

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25 views

### Construction of point process having same pair correlations as GUE

The distribution of the pair correlations of the eigenvalues of the GUE satisfies (in the limit, when being normalized appropriately)
$$
g(u) = 1 - \left(\frac{\sin(\pi u)}{\pi u}\right)^2 + \delta(u)....

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103 views

### Weighted global Holder property for Brownian motion paths

It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1
$$
\sup_{t,s\in[0,1]}\frac{|W_t-W_s|}{|t-s|^{\...

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146 views

### Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...

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68 views

### Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...

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39 views

### Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...

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78 views

### Brownian motion - probability of hitting an open subset of the sphere

Consider a Brownian particle in $\mathbb{R}^n$, starting at the origin. Let $\mathbb{P}_t(A)$ be the probability of the particle striking $A \subset S^{n - 1}$ within time $t$, where $A = \{ (x_1, x_2,...

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53 views

### Some problems about symmetric convolution semigroup on the unit circle

These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes".
Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...

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129 views

### Transition probabilities for the symmetric random walk on the integers

I found that most references for the symmetric random walk on the integers are for the discrete time case, i.e. the ones that gives us explicit transition probabilities. Now, I am looking at a random ...

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277 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 = \...

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141 views

### Linking Wasserstein and total variation distances

I seek to bound the total-variation distance between two probability measures $p_1$ and $p_2$. It is extremely easy to build a parameter space where $p_1$ and $p_2$ are the marginals of some joint ...

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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 ...

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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, ...

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56 views

### Circular process ergodic?

Let us define a continuous-time Markov process on a circle consisting of $m-$ equally spaced points, i.e. every point has two neighbours.
Now, we define a space of functions $S:= \{-1,1\}^{\{1,...,m\}...

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86 views

### Weak convergence of process

Background:
I am trying to compute the weak limit of the following model from mathematical biology that is supposed to exist:
Let $$L(f)(\eta)= \sum_{x \in \mathbb{Z}}\frac{1}{2}\left(1_{\eta(x+1) \...

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91 views

### Malliavin differentiability of solutions to SDEs

In Bass's book on Diffusions and Elliptic Operators, the author gives a brief introduction into Malliavin Calculus. He calls a functional $F:C([0,1],\mathbb{R})\rightarrow \mathbb{R}$ $L^p-$smooth if ...

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112 views

### Carre du Champ, Subunit Paths and CC-metrics

Let the operator $L$ be given by $Lf(x):=\nabla\cdot (A\nabla f(x))$, where $f:\mathbb{R}^d\rightarrow \mathbb{R}$ belongs to a suitable class of functions $\mathcal{A}$. The carre du champ operator $\...

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69 views

### Definition: Grigelionis Process?ch [closed]

Background
I've been reading this article and it keeps referring to "Grigelionis processes", which apparently generalize Levy processes. However the paper does not define these object clearly and ...

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126 views

### Malliavin derivative under change of measure

Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$.
Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative ...

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66 views

### Quadratic variation and the variance of a semimartingales

I will describe an example that seemingly contradicts the following
Theorem
For a local martingale $M$, let $[M,M]_t$ be its quadratic variation at $t$. For any $t$, if $E[[M,M]_t]<\infty$, then $...

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48 views

### Regularity of the entrance measure of SRW

Let $S(n)$ be the discrete sphere of radius $n$ (i.e., the internal boundary of the Euclidean discrete ball $B(n)$) centered in the origin, and consider a simple random walk starting at some $x\in\...

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100 views

### Is a local martingale with constant expectation necessarily a martingale?

Suppose $X\in \mathbb R$ is a weak solution to the SDE $dX_t = \sigma(X_t)dW_t$, in which $W$ is a one-dimensional Brownian motion, and $\sigma$ is Borel measurable so that a weak solution exists and ...

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176 views

### Brownian motion in $\mathbb{R}^n$, probability of hitting a set

Consider a particle undergoing Brownian motion in $\mathbb{R}^n$, starting at the origin, and let $B(t)$ denote its position at time $t$. Let $X$ be an arbitrary subset of $\mathbb{R}^n$. I am trying ...

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100 views

### Example of progressively measurable process that is not predictable

Is there an example of progressively measurable process that is not predictable?
This question is motivated by Revuz-Yor, Continuous Martingales and Brownian Motion http://www.springer.com/gb/book/...

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68 views

### Order statistic of Markov chain sample path and related probabilities

Consider a 1D sample path, denoted as $\{X(1), ..., X(t), ..., X(n)\}$, generated from a discrete time finite state (time homogeneous) Markov chain over states $\{1,...,m\}$, with transition ...

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142 views

### Total absolute variation of brownian motion, with different sampling rates

Let $(B_t)$ be a brownian motion on [0,1]. For the following, let $\omega$ be fixed.
Let's compute the total absolute variation when sampling period = $\delta$ is fixed:
$$V(\delta) = \sum_{i=0}^{N-...

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31 views

### Comparison between the entrance measure and the harmonic measure

Consider the standard two-dimensional Brownian motion, and define $\tau(A)$ to be the hitting time of $A\subset \mathbb{R}^2$. Let $hm_A$ be the harmonic measure (from infinity) on $A$. Let $B(r)$ be ...

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123 views

### Malliavin Calculus: directional derivatives of cylinder functions exist in what sense?

Denote by $P_0(\mathbb{R}^d)$ the sets of continuous paths over $[0,1]$ started at $x=0$ with values in $\mathbb{R}^d$, we equip this space with the sup-norm and make it into a probability space by ...

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58 views

### “Convergence speed” results for the Langevin process

The Langevin process is defined by the following stochastic differential equation:
$$ \dot X = - \nabla \phi + \sqrt 2 dW_t $$
Its equilibrium distribution is the following:
$$ p_\infty (x) \propto ...

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99 views

### Full version of Soucaliuc's research announcement “Réflexion entre deux diffusions conjuguées”

Florin Soucaliuc published the following research announcement in 2002 containing some results from his thesis on reflected diffusion processes:
[1] F. Soucaliuc, Réflexion entre deux diffusions ...

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113 views

### Laplace transform of a integral function of CIR/CEV process

The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...

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110 views

### Feller processes / probability generators

I am looking for a example of a function in $C_0(\mathbb{R})$ such that $f',f'' \,\text{and}\, f''' \in C_0(\mathbb{R})$ with
$$ \inf f < \inf (f-a*f''')$$ for some $a>0$, but I couldn't find ...

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109 views

### Comparison of Parameter estimation using maximum likelihood and Maximum entropy

I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...

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33 views

### steady state distribution for a jump Markov chain

Consider a queueing process with the following transition matrix:
$\mathbf{P}=\left( \begin{smallmatrix}
1-\lambda & \lambda & & & & & & &\\
\mu & 1-\...

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175 views

### Random walk with continuously distributed steps on [-1,1]

A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability
$$P(S_n \textrm{ reaches } a \textrm{ before} -b) =...

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88 views

### Finite hitting time implies hits at any finite time?

I was wondering about the following problem:
Assume we have a state space $S:=\mathbb{Z}$ and a Markov chain, such that we can go from any state $x$ to some state $y$ with positive probabilities, i.e....

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87 views

### Weak existence for modified Tanaka SDE

Tanaka's theorem (wikipedia) implies that $X_t = |B_t|$ is a weak solution to the SDE
$dX_t = dW_t + dL_t^0(X_t)$,
where $W_t$ is a Brownian motion and $L_t^0(X_t)$ is the local time of $X_t$ at $0$....

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111 views

### Markov chain with Feller property

Does anybody know whether there is an analysis of when the monotone decreasing chain has the Feller-property?
The monotone decreasing is defined as a chain on $\mathbb{N}$ and the rate of going down $...

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53 views

### hitting time of an Ornstein-Ulhenbeck

If we consider a nice Ornstein Uhlenbeck process
$d x (t) = - \gamma x(t) dt + \sigma d w (t)$
with $x(0) = x_0 \in (-L,L)$.
Here $\gamma, \sigma$ are positive constant and $w(t)$ is a Wiener process....

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157 views

### approximate stationary distributions of a doubly stochastic matrix and its supports

Given a doubly stochastic matrix $M$ and a distribution $v$,let $M=\sum_{\sigma\in S_n}p_{\sigma}M_{\sigma}$ be any Birkhoff decomposition of $M$, where $M_{\sigma}$ is the permutation matrix induced ...

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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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59 views

### Reference for branching processes

A popular model of a continuous time branching process was introduced around 1970, which is now called the Crump-Mode-Jagers branching process, was introduced here:
A General Age-Dependent Branching ...

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162 views

### using Feynman-Kac formula

I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the ...

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### 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),...

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67 views

### Random process & probability problem met in wireless communication

A random process r obeys the following distribution:
$p(r,ṙ)=rb_0\exp(−\frac{r^2}{2b_0})\sqrt{2πb_2}exp(−\frac{\dot{r}^2}{2b_2})$, where $\dot{r}$ is the derivative of r in the time domain.
You can ...

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192 views

### General solution to system of stochastic linear differential equations

Assume we are given the system of linear stochastic differential equations
$$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot dt+\...

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19 views

### Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup.
Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let $...

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117 views

### Onsager-Machlup function for special matrix-valued diffusion process

Potentially useful background info
For standard vector-valued diffusion processes the following result is well-known:
Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by
\begin{align*}
...

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215 views

### Intuition about Skorohod integral

I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on.
In particular ...