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

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131 views

### $\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales?

I am trying to prove the following Lemma, which seems intuitive, but I still have doubts:
Lemma
Given a Brownian motion $\{W_t,\mathcal F_t:0\le t \le1\}$, two bounded processes, $\mu$ and $\sigma$, ...

**5**

votes

**2**answers

259 views

### $L^\infty$ properties of an infinite-dimensional Gaussian semigroup

Let $W$ be a separable Banach space and $\mu$ a Gaussian Borel measure on $W$ which is centered and non-degenerate. For $F : W \to \mathbb{R}$ bounded Borel and $t \ge 0$, let
$$P_t F(x) = \int_W ...

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vote

**1**answer

126 views

### Time integral of a diffusion

Define $\bar\sigma^2_t=\frac{1}{t}\int_0^t\sigma^2(X_s)ds$ where $\sigma(x)\geq0$ is a measurable function and $X_t$ a diffusion process defined by
\begin{equation}
...

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**1**answer

223 views

### Convergence of random variables with hypergeometric distribution

This is a very interesting conjecture of large scale property of hypergeometric distribution.
Let $a>1$ be a integer constant, $N\in\mathbb{N_+}$, for any $x<N-1$, consider $N+(a-1)x$ balls in ...

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**1**answer

322 views

### question about uniform continuity under Skorokhod Metric

Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see:
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g
...

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votes

**0**answers

62 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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42 views

### How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...

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**1**answer

90 views

### Question about the stochastic integral of martingales

Let $M=(M_t)_{t\ge 0}$ be a continuous martingale defined on some filtered probability space taking values in $R$. Let $H=(H_t)_{t\ge 0}$ be some bounded progressively measurable process, i.e.
...

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votes

**1**answer

1k views

### Gluing Markov processes

I am looking for a reference on the gluing together of strong Markov processes to get a new one.
Here is an example of what I have in mind. Let $B^1, B^2, \ldots $ be independent one-dimensional ...

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**1**answer

388 views

### Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail.
Here is what I mean exactly. ...

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votes

**1**answer

214 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 ...

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votes

**0**answers

79 views

### When an integral with respect to a Poisson point process is finite?

Let $N(ds,dv)$ be a Poisson measure on $\mathbb{R} _+ \times \mathbb{R} _+$ with intensity $dsdv$. Let $N = \sum\limits \delta_{(s_i,v_i)}$. Assume that $N$ is compatible with a filtration $\{ ...

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**1**answer

145 views

### explicit characterization of the stochastic integrand

Let $V$ be a cadlag positive supermartingale with the following decomposition:
$$V_t=V_0+\int_0^tH_sdX_s-K_t$$
where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with ...

**2**

votes

**0**answers

43 views

### probabilistic interpretation of elliptic equation with mixed boundary condition

I would like to understand the probabilistic interpretation of the following elliptic problem with mixed Dirichlet-Neumann boundary conditions:
Let $B := \{ x \in \mathbb{R}^n, \quad \| x \|_2 \leq 1 ...

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**0**answers

67 views

### Rolling map as a diffeomorphism?

Let $M$ be a (compact) Riemannian manifold and $x \in M$. For a piecewise smooth path $\gamma: [0, T] \longrightarrow M$, we can define Cartan's development map (or rolling map)
$$(\Phi\gamma)(t) = ...

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**1**answer

203 views

### Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution?
$$
\int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau}
$$
where ...

**4**

votes

**1**answer

312 views

### Law of Iterated Logarithm for autoregressive process

Suppose that $\{X_i\}$ is an $\mathrm{AR}(r)$, defined by:
$X_{i}= h(i) + \varepsilon_i $,
$h(i)=\alpha_1 X_{i-1} + \dots + \alpha_{r} X_{i-r}$
where $\{\varepsilon_i\}$ are i.i.d. ${\cal ...

**4**

votes

**1**answer

87 views

### Relative vulnerabilities in SIS epidemic model

Consider the SIS model of epidemic spreading. There is a finite graph $G(V,E)$, link infection rates $\lambda_{ij}$ and node recovery rates $\mu_i$. There are a few initial nodes which are infected at ...

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**1**answer

196 views

### Quasi-stationary distribution for a death process

In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived.
Consider a ...

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**1**answer

335 views

### Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...

**0**

votes

**1**answer

176 views

### Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...

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votes

**0**answers

52 views

### Consistency Conditions of the Kolmogorov Extension Theorem

Kolmogorov's extension theorem allows for the construction of a variety of measures on infinite-dimensional spaces, and its conditions are supposedly "trivially satisfied by any stochastic process". ...

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**0**answers

27 views

### References for symmetric α-stable process (SSP) for $a>2$

Many properties of Brownian motion have been extended to SSP's for $0\leq \alpha\leq 2$ and so it is quite easy to find literature on them. However, I am currently studying the SSP for $\alpha>2$ ...

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votes

**1**answer

216 views

### weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...

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**0**answers

81 views

### Equivalence of two non-degenerate Gaussian measures on Banach space

The motivation of this question is to show that two probabilities on
$C_{0}^{n}(0,1)$ (the space of continuous $\mathbb R^{n}$ valued process
on $[0,1]$ starting from zero) induced by two ...

**2**

votes

**1**answer

121 views

### What are the generalized Gaussian probability laws that are infinitely divisible?

We consider the probability density, often called a generalized Gaussian density, $$p_{\alpha}(t) \propto \exp (- |t|^\alpha),$$
with parameter $0<\alpha<\infty$. For $p = 2$, we recognize a ...

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**0**answers

34 views

### Killing a Feller Process

Given a canonical Feller process $(X_t,P_x)$ with Feller semigroup $P$. Let $T$ a (good) stopping time, for example $T=\inf\{u\ge 0 : X_u=0\}$. I'm looking for a proof of the following claim
...

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**2**answers

89 views

### Version of Ito's lemma applied to a stochastic function

The Ito's formula stated in most books in stochastic calculus is in the form $F(t,X_t)$, where $F: \mathbb{R}^{d+1} \rightarrow \mathbb{R}$ is a $d+1-$dimensional deterministic $C^{1,2}$ function and ...

**2**

votes

**2**answers

165 views

### Uniqueness in martingale representation theorem

Dudley's martingale representation theorem states that if $W=\{W_t,\mathcal{F}_t;0\le t<+\infty\}$ is a standard one-dimensional Brownian motion, $0<T<+\infty$ and $\xi$ is ...

**1**

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**1**answer

55 views

### Correlation between two continuous-time stochastic processes

Consider two continous-time stochastic processes $\{A(t)\}_{t \ge 0}$ and $\{B(t)\}_{t \ge 0}$ with $A(t)=t$ and $B(t)=t$. Each process starts at $t=0$ and emits "ticks" at increasing time slots. For ...

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**2**answers

104 views

### Origins and Industrial Applications of stochastic processes (eg. Brownian motion) on Riemannian manifolds

I am studying BM on Riemannian manifolds and I am curious how this theory started. In the references below (esp. in Hsu's exposition), you will find many applications of that theory such as a ...

**2**

votes

**0**answers

72 views

### Convergence in distribution of random measures

Let $M$ denote the space of real Radon measures on $\mathbb{R}$ as the topological dual of $C_c(\mathbb{R})$ equipped with the inductive limit topology (for possibly unbounded Radon measures) or ...

**3**

votes

**3**answers

311 views

### Reference request: a guide through quantum probability

Could you point out a comprehensive reference book (or more than one, if it is the case) on Quantum Probability that introduces the subject and then gradually builds up to the edges of contemporary ...

**1**

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**1**answer

149 views

### Diffusion processes with different diffusion coefficients and absolute continuity

I would first of all like to say that I am an analyst, and so I am familiar with probabilistic methods only on a basic level.
My initial situation is the following. Consider two stochastic ...

**3**

votes

**0**answers

88 views

### On the decay of correlations of an ergodic sequence over the set $X_{0}=0$

The following question arose while I was trying to explore possible further extensions of a CLT by Liverani which I mentioned here already (see this link, I can tell you more details upon request). It ...

**1**

vote

**1**answer

581 views

### Ergodicity of a Markov chain

Hi,
I'd appreciate some help on a Markov chain result I'm trying to show. I believe the following is sufficient for a continuous time Markov chain $(X_t)$ with a countable state space to be ergodic:
...

**0**

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**0**answers

30 views

### Reference request for specific POMDP examples

Following is strictly for discrete-time discrete-space Markov chain.
Consider a partially observed Markov decision process (POMDP) $P = \{X,O,A,P,B_a\}$.
Here $X = \{x_1, \cdots, x_n\}$ refers to ...

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**2**answers

929 views

### Derivative of a differentiable stationary Gaussian process

Thanks for your help in advance. I'm interested in understanding the properties of derivatives of a differentiable stationary Gaussian process. Specifically, is the derivative also a Gaussian ...

**4**

votes

**1**answer

82 views

### On Minkowski sum of two independent Poisson point processes

Suppose that $\Phi_1$ and $\Phi_2$ represents two independent Poisson point processes respectively with intensity $\lambda_1$ and $\lambda_2$ (therefore). We know very well different operations on ...

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**0**answers

34 views

### Progressive measurability and functional composition

Suppose we have a progressively measurable process $X$ taking values in $\mathbb{R}^d$. What are sufficient conditions on a function $f( x, t, \omega ) \colon \mathbb{R}^d \times [0,\infty) \times ...

**0**

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**0**answers

69 views

### Uniqueness of a Integro-parabolic differential equation?

Let $r, q,\lambda,\sigma,\kappa,\mu$ are positive real numbers and let $c(t)$ is a differential function of $t$. $\Gamma(\eta)$ is a probability density function.
When I consider price of American ...

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**0**answers

74 views

### Convergence in distribution of stochastic equation solutions

I post this post en MSE (link) but I think that is more suitable for this site.
I'm studying from Kurtz's book "Markov Processes Characterization and convergence" and I have a question about the ...

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**0**answers

39 views

### Definition of mth order stationarity

in the definition of the weak GARCH processes they use the terminology of the 4th-order stationarity of the process $(X_t)$. I know the definition of 2n-order stationarity, but I'm not exactly sure, ...

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76 views

### Stationarity of Brownian motion with drift

Suppose the following SDE for $X_t$ is well-posed:
$$dX_t = \sqrt{2}\, dB_t - \nabla\Phi(X_t)\,dt.$$
For what $\Phi\in C^1(R^d)$ will $X$ have stationary distribution $u_{\infty}$? For what $\Phi$ ...

**0**

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**0**answers

41 views

### Integral over a point process. Asymptotic of the dispersion

I consider an integral (or a sum with random index)
$$
M(t) =\int\limits_0^t f(t-u)dX(u),
$$
where
$$
X(u) = \sum\limits_{i=1}^{N(u)} \xi_i,\qquad N(u)=\max\{k: \tau_1+\,\dots,\,\tau_k\, <\, u\},
...

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**0**answers

25 views

### Moments in the Quantile Process

Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F:
$$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$
In this case, $F^{-1}$ is the (generalized) inverse of ...

**2**

votes

**1**answer

119 views

### Ergodicity for the mean of a linear process without finite second moment

Suppose that $\{X_k:k\in\mathbb Z\}$ is a linear process, i.e. a sequence of random variables such that
$$
X_k=\sum_{j=0}^\infty\psi_j\varepsilon_{k-j}
$$
for each $k\in\mathbb Z$, where ...

**0**

votes

**0**answers

33 views

### Bahadur-Kiefer representation and KMT embedding

I am interested in the connection between the so called Bahadur-Kiefer process and the KMT/Hungarian embedding. At first sight there seems to be a relationship between the topics, but oddly enough ...

**4**

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**1**answer

190 views

### Stability of convergence in distribution under randomization

Suppose you have a sequence of non-negative stochastic processes $(X^n)_{t \in \mathbb{R}}$, $n \geq 1$, with continuous paths and continuous in $t$ such that
$$\int_{-\infty}^{\infty} X^n_t \, ...

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**2**answers

477 views

### Converse to Girsanov's theorem?

Roughly speaking, Girsanov's theorem says that if we have a Brownian motion $W$ on $[0,T]$, we can construct a new process with a modified drift that has an equivalent law to $W$ (subject to ...