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

**4**

votes

**1**answer

70 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 ...

**0**

votes

**0**answers

34 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 ...

**2**

votes

**1**answer

134 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 ...

**5**

votes

**0**answers

70 views

### An Interesting Markov Chain Mixing time question (coupling of random variables)

Consider a simple markov chain called reproduction process. Let $N>0$ be fixed. At each time $t$, there are $N$ balls, in which $x$ balls are colored white, and $N-x$ balls are colored black. We ...

**1**

vote

**0**answers

23 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**

votes

**0**answers

24 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.
...

**0**

votes

**1**answer

62 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

**0**answers

61 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

290 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 ...

**0**

votes

**0**answers

33 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, ...

**3**

votes

**0**answers

58 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 $\Phi$ ...

**4**

votes

**1**answer

73 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 ...

**0**

votes

**0**answers

36 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\},
...

**2**

votes

**1**answer

183 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 ...

**0**

votes

**0**answers

22 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 ...

**5**

votes

**1**answer

300 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 ...

**2**

votes

**1**answer

106 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

**1**answer

149 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 ...

**-1**

votes

**0**answers

74 views

### Stochastic ordering preserving operators on $n$ random variables

This question is about the operators which can preserve stochastic ordering.
Let $\mathcal{L}:\mathbf{X}\rightarrow \mathbb{R}$ be an operator where $\mathbf{X}=[X_1,X_2,...,X_n]$ and consider two ...

**2**

votes

**1**answer

203 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 $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...

**0**

votes

**0**answers

29 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**

votes

**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 \, ...

**4**

votes

**2**answers

462 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 ...

**2**

votes

**1**answer

90 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

52 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 ...

**1**

vote

**1**answer

134 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 ...

**2**

votes

**1**answer

203 views

### A calculation involving a uniform random variable quantile

THE PROBLEM:
Let $U$ be a uniform distribution and $U_{n}$ be its nth empirical distribution. Suppose $t\in (0,1)$ and $n\in \mathbb{N}$ are constants. What's the explicit expression to
...

**1**

vote

**1**answer

273 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
...

**7**

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 ...

**3**

votes

**1**answer

69 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 $\langle M\rangle=\langle N\rangle$, ...

**6**

votes

**1**answer

343 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. ...

**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 ...

**0**

votes

**1**answer

199 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 ...

**6**

votes

**2**answers

232 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 ...

**0**

votes

**0**answers

34 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 ...

**2**

votes

**0**answers

60 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

43 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 ...

**8**

votes

**1**answer

433 views

### Joint law of the time integral of Brownian motion and its maximum

Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively:
$$M_t=\max_{0\leq s\leq t}\,W_s$$
...

**3**

votes

**1**answer

95 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

98 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

92 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

120 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 ...

**2**

votes

**0**answers

51 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**

vote

**1**answer

122 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 ...

**-1**

votes

**1**answer

114 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

41 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

172 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

59 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

268 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

27 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 ...