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

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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) = ...

**0**

votes

**0**answers

77 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 $\{ ...

**2**

votes

**0**answers

40 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 ...

**0**

votes

**0**answers

78 views

### Proof of $\lim_{t\rightarrow 0} \mathbb E f(S_t)=f(0)$ for a diffusion $S_t$?

I am trying to prove the following statement for a diffusion $S_t$ with $S_0=0$ and a real function $f$ that is continuous at $0$:
$$\lim_{t\rightarrow 0} \mathbb E f(S_t) = f(0), \text{ if } \mathbb ...

**0**

votes

**0**answers

53 views

### Derivative of the Expectation of an Integral over a Diffusion

I am trying to prove the following, where $S_t$ is a diffusion:
$$ \lim_{t\rightarrow 0}\frac 1 t \mathbb E \int_0^t f(S_s)ds = \lim_{t\rightarrow 0} \mathbb E f(S_t) $$
Proof attempt:
Lusin's ...

**0**

votes

**0**answers

51 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". ...

**0**

votes

**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 $\alpha>2$ ...

**2**

votes

**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

**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
...

**0**

votes

**2**answers

86 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 ...

**5**

votes

**2**answers

102 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 ...

**1**

vote

**1**answer

54 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 ...

**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

**0**answers

85 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 ...

**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 ...

**0**

votes

**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 ...

**3**

votes

**1**answer

211 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 ...

**0**

votes

**1**answer

195 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 ...

**0**

votes

**1**answer

75 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.
...

**1**

vote

**0**answers

33 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 ...

**2**

votes

**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 ...

**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 ...

**3**

votes

**0**answers

74 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 $\Phi$ ...

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votes

**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, ...

**0**

votes

**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\},
...

**0**

votes

**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 ...

**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 ...

**2**

votes

**1**answer

118 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

68 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 ...

**2**

votes

**2**answers

162 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**

vote

**0**answers

56 views

### Stochastic Resonance in Infinite Dimensions

I'll ask this from the point of view of physics more than of theoretical mathematics. I'm searching for a mathematical discussion of stochastic resonance interpreted in a PDE sense. This is a good ...

**0**

votes

**0**answers

23 views

### Density of $\int_{B}\frac{|1-|B_{T}|^{2}|}{|y-B_{T}|^{3}}dS(y)$

For $B\subset \partial B(0,1)))$ and random variable $B_{T}\in Int(B(0,1))$ with density $p_{T}$, is there a density for
$\int_{B}\frac{|1-|B_{T}|^{2}|}{|y-B_{T}|^{3}}dS(y)$?
Context
The original ...

**3**

votes

**1**answer

74 views

### Could quadratic variation determine distribution?

Let $M=\{M_t,\mathcal{F}_t;0\le t<+\infty\}$, $N=\{N_t,\mathcal{F}_t;0\le t<+\infty\}$ be two continuous local martingales with $M_0=N_0=0\text{ a.s.}$. If $\langle M\rangle=\langle N\rangle$, ...

**0**

votes

**0**answers

37 views

### Finding a stochastic differential equation as limit of a discrete stochastic equation

I'm dealing with the following problem:
Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation:
$Z_{k+1}-Z_k=P_k(1-2Z_k)$
where $P_k=0$ with probability ...

**6**

votes

**2**answers

254 views

### A version of Wald identity

Let $W$ be a standard one-dimensional Brownian motion. Let $T$ be a stopping time with $\mathbb{E}\sqrt{T}<+\infty$. Then
$$\mathbb{E}W_T=0\quad \mathbb{E}W^2_T=\mathbb{E}T$$
I can prove these ...

**2**

votes

**0**answers

61 views

### Existence of 1-1 mapping/homeomorphism

Let $B$ be a standard 2-D Brownian motion, and $\sigma: \Omega\times \mathbb R^{+} \mapsto \mathbb R^{2 \times 2}$ is an $\mathcal F_{t}$ adapted process satisfying, for some constants ...

**2**

votes

**0**answers

47 views

### What is the probability of B.M. hitting two disjoint spheres $(d\geq 3)$?

The hitting probability for spheres centered at origin is $P_{x}(T_{B_{r}(0)}<\infty)=\frac{r^{d-2}}{|x|^{d-2}}>0$, where $|x|>r$.
1)So I was wondering how can one compute ...

**3**

votes

**1**answer

107 views

### Do we need Feller condition if the process jumps?

Consider the SDE:
\begin{equation}
dv_t = k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{v}_{t}
\end{equation}
It describes a process $v_t$ which is a strictly positive if the drift is stronger enough, i.e. ...

**1**

vote

**0**answers

109 views

### How to show that two linear combinations of Bernoulli random variables have jointly Gaussian distribution (and more)

Let $X_1,\ldots,X_n$ be independent Bernoulli random variables such that $\mathbb{P}(X_i=\pm 1)=1/2$ and consider two collections of real numbers $a_1,\ldots,a_n, b_1,\ldots, b_n$. For the moment let ...

**2**

votes

**0**answers

96 views

### Speed of Approach to Invariant Measure

Let $X_t$ represent a continuous-time Markov process on $\mathbb{R}^d$, say a diffusion with locally Lipschitz coefficients. Suppose that there exists a unique invariant measure $\mu$ on the space, ...

**2**

votes

**1**answer

129 views

### Onsager-Machlup function and most probable path of a diffusion process

Let $X_{t}$ be a real, one-dimensional diffusion process satisfying the stochastic differential equation
\begin{equation}
dX_{t} = f(X_{t})dt + dW_{t},
\end{equation}
where $f \in C_{b}^{2}(R)$ is a ...

**1**

vote

**1**answer

130 views

### Does very fast convergence in probability imply almost sur convergence for a continuous stochastic process?

I was wondering if someone knows how to prove the following fact (which might not be a fact ;) ):
let X being a stochastic process with almost surely continuous sample path, and such that, there ...

**2**

votes

**0**answers

61 views

### Property of relative entropy [closed]

For $X$ a measurable space and $P,Q$ two probability measure on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as
$$D(Q\|P)=\int_X ...

**-1**

votes

**1**answer

115 views

### Property of relative entropy [closed]

For $X$ a measurable space and $P,Q$ two probability measures on $X$ s.t. $Q$ is absolutely continuous with respect to $P$, the relative entropy is defined as
$$D(Q\|P)=\int_X \log(\frac{dQ}{dP})dQ,$$
...

**2**

votes

**0**answers

43 views

### Local time for drifted Brownian motion and comparison results for reflected diffusion

Suppose $X(t) = x+ \mu t + \sigma W(t)$ where $x\ge 0$, $\mu, \sigma>0$ are real constants, and $W$ is a standard Brownian motion. The Skorohod decomposition of $X(t)$ can be written as $Z(t) = ...

**5**

votes

**0**answers

184 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 ...

**3**

votes

**0**answers

60 views

### Lorenz attractor power spectrum

If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...

**10**

votes

**1**answer

269 views

### Does Brownian motion immediately visit both sides of a Jordan curve?

Let $C$ be a Jordan curve in $\mathbb{R}^2$. By the Jordan curve theorem, $\mathbb{R}^2 \smallsetminus C$ is uniquely partitioned into two connected regions $A$ and $B$ (the interior and exterior).
...

**0**

votes

**0**answers

28 views

### sign and absolute value at fixed time of a diffusion process

I have a diffusion in the plane $(X,Y)$ with Feller semigroup such that each coordinate is a standard Brownian motion, $|X|=|Y|$ and $(X,Y), (Y,X)$ have the same law. I want to prove that for a ...

**2**

votes

**0**answers

67 views

### Sum of the entries of the inverse covariance matrix

Let $T \in\left(0,1\right)$, $n\in\mathbb{N}$ and $e_n = [1,\ldots,1]\in\mathbb{R}^n$. Consider the covariance matrix $\mathfrak{A}_n = ...