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

**4**

votes

**1**answer

89 views

### On the moments of Lévy processes

For a Brownian motion $B_t$, the evolution of the moments with $t$ obeys the simple rule:
$$\mathbb{E}[|B_t|^p] = \kappa_p |t|^{p/2},$$
with $\kappa_p<\infty$. The proof only requires to remark ...

**4**

votes

**1**answer

180 views

### Concurrency related problems in $n$ independent, parallel $M/M/1$ queues

Queueing Model:
Consider $n$ independent, parallel $M/M/1$ queues with identical arrival rate $\lambda$ and service rate $\mu$. For each $M/M/1$ queue, we use the FCFS (First Come First Served) ...

**1**

vote

**0**answers

189 views

### Fundamental theorem of calculus for iterated stochastic integrals

I'm trying to find the rate (or a bound for it) with which an iterated integral of the type
$$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$$
converges to zero (in probability/distribution) for $h ...

**6**

votes

**1**answer

703 views

### Properties of the time integral of Wiener process

Let $W_t$ be a Wiener process and consider the time integral
$$ X_T:= \int_0^T W_t dt $$
It is often mentionend in literature that $X_T$ is a Gaussian
with mean 0 and variance $T^3/6$.
I am ...

**7**

votes

**1**answer

329 views

### About the convergence rate for an approximation to the heat kernel

Let $G(t,x)$ be the heat kernel
$$
G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}.
$$
Here is one approximation to $G(t,x)$:
$$
G_\epsilon(t,x)=e^{-t/\epsilon} ...

**1**

vote

**0**answers

48 views

### A counterpart of Karhunen theorem

According to the Karhunen theorem, if the correlation function of a process $X(t)$
can be represented as
$$
R(t,s)= \int_{\Lambda} f(t, \lambda) \overline{f(s, \lambda)}d\nu(\lambda)
$$
then the ...

**4**

votes

**3**answers

321 views

### Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation
\begin{equation}
dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0,
\end{equation}
where ...

**1**

vote

**1**answer

138 views

### M/M/1 Queue with probability of new customer leaving [closed]

I'm looking at a M/M/1 queue system and trying to show that $\{M_t\}_{t\geq}0$, the number of clients in the system, is a birth-death process. In the simplest of cases this is true if $\lambda_i = ...

**1**

vote

**0**answers

114 views

### number of times Brownian motion hits boundaries

Any experts here please direct me to some appropriate keywords that I can search for. Consider a Brownian motion constrained to an upper and lower boundaries. Let's say I want to know that how many ...

**2**

votes

**3**answers

73 views

### a special filtration satisfying $0$-$1$ law

Let $\xi$ be a uniformly random variable on $[0,1]$ defined on some probability space $(\Omega,\mathcal{F})$. Define the process $\xi_t:=\min(\xi,t)$ for $0\le t\le 1$. And let ...

**3**

votes

**1**answer

197 views

### Quantiles moments and Convergence

QUESTION:
Let $F$ be an absolutely continuous distribution function with density $f$, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence
...

**2**

votes

**1**answer

218 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

364 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
...

**0**

votes

**1**answer

71 views

### Running supremmum of a Levy process

Let X be a cadlag Lévy process with $X_0=0$ and let $p$ be a real number in $[1,\infty)$. Then, the following are equivalent.
1): $X$ is $L^p$-integrable.
2): $X^*_t= \mathop{\sup}_{0\leq s\leq t} ...

**1**

vote

**0**answers

42 views

### a question about the modification of a supermartingale

Let $\mathbf{D}\subset\mathbf{D}([0,1],\mathbb{R}_+)$ denote the space of positive cadlag functions $\mathbf{x}$ defined on $[0,1]$ with $\mathbf{x}(0)=1$. Define the canonical process
...

**2**

votes

**1**answer

167 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

**1**answer

189 views

### multiplication of two ergodic and stationary processes

If X and Y are stationary and ergodic processes, then, is XY a stationary and ergodic process?
I think the answer is true, but I do not know how to find the mean (we do not know Y and X are ...

**3**

votes

**2**answers

301 views

### Convergence of iterated stochastic matrices

It is well-known that for a stochastic aperiodic matrix $M$,
the sequence $(M^n)_n$ converges.
Here I would like to a have a more precise analysis. Consider now a sequence of stochastic matrices ...

**5**

votes

**0**answers

337 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
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...

**1**

vote

**0**answers

90 views

### question about Doob-Meyer decomposition

Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition:
...

**5**

votes

**2**answers

120 views

### Smoothness of $g(t,x)=\mathbb{E}[f(X_T)|\mathcal{F}_t]$

Assume a process with Itô dynamics of the generic form
$$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t$$
and let $f:\mathbb{R}\to\mathbb{R}$ be borel-measurable. Is the following function smooth ?
...

**1**

vote

**0**answers

106 views

### Arctic Circle Theorems and the Wave Equation

I've seen the following remark in a number of papers but don't know what to make of it. In this paper by Cohn, Elkies and Propp, it is mentioned that the normalized average Height function ...

**1**

vote

**1**answer

337 views

### Strong solutions on SDE (stochastic differential equations) with discontinuous drift and diffusion coefficients

I want to get some advice from you about the existence (and the uniqueness if possible) of a strong solution on my SDE. In fact, due to the structure of the problem that I consider, both the drift ...

**2**

votes

**0**answers

98 views

### Stationary Distribution for Markov-like system?

Let
\begin{equation}
A=
\begin{pmatrix}
0 & a_{1,2} & a_{1,3} \\
a_{2,1} & 0 & a_{2,3} \\
a_{3,1} & a_{3,2} & 0
\end{pmatrix},
\end{equation}
\begin{equation}
B=
...

**8**

votes

**3**answers

547 views

### Maximum of the expectation of maximum of Gaussian variables

Suppose $X=(X_1,\ldots,X_n)$ is a Gaussian vector with each entry $X_i$ marginally distributed as $\mathcal{N}(0,1)$. Want to find out the possible maximum of
$$\mathbb{E}\max_{1\le i\le n}|X_i|$$
and
...

**2**

votes

**0**answers

92 views

### Learning resources for Probability Distributions/Models [closed]

I've a good background in basic probability. I need to learn and get a good grip on the probability distributions and stochastic processes, counting processes, and other related topics.
I am already ...

**1**

vote

**0**answers

86 views

### asymptotic estimate of random walk involved hitting time and return time

Consider a reversible random walk on (say) $\mathbb{Z}$, are there any estimate for the following probability $\mathbb{P}(\tau_n=m<\tau_0^+)$ where $\tau_n$ is the first hitting time at site n and ...

**2**

votes

**0**answers

81 views

### Existence of a conditional distribution

Let $X$ and $Y$ be standard Borel spaces and let $J$ be an analytic subset of $X\times \mathcal P(Y)$ where $\mathcal P(\Omega)$ is a set of probability measures on a Borel space $\Omega$ endowed ...

**3**

votes

**0**answers

96 views

### Nonlinear Markov process

Consider the following nonlinear $\mathbb{R}$-valued stochastic recursive sequence:
$ X_{n+1} = F(X_n) + W_{n+1}, \quad (W_n)_{n\ge1} \stackrel{ \scriptsize \mathrm{i.i.d.} }{ \sim } \phi. $
How can ...

**13**

votes

**1**answer

401 views

### Fictitious density of paths of diffusion processes outside the Cameron--Martin space

Let $X_t$ be an $n$-dimensional diffusion process satisfying the following Itō SDE over $[0,1]$:
$$dX_t = f(X_t)\,dt + dW_t,$$
where $W_t$ is an $n$-dimensional Wiener process and $f$ is of class ...

**4**

votes

**0**answers

58 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 ...

**2**

votes

**1**answer

72 views

### Why does differencing create wide-sense stationary time series?

In time series analysis, a common assumption made is that the series is wide-sense stationary, ex. that it has time invariant mean and covariance. However, as this is often not the case in real life, ...

**1**

vote

**1**answer

168 views

### On the solution of a stochastic partial differential equation

Consider a simple SPDE as follows:
$\partial_t u(t,x)=\partial_x^2 u(t,x)+V(u(t,x))+\dot{W}(t,x)$, $t>0$, $x\in(0,1)$,
$u(t,0)=u(t,1)=0$,
$u(0,x)=v(x)$,
where $V$ is a bounded, smooth ...

**5**

votes

**1**answer

164 views

### Escape Time of Fractional Brownian Motion

Let $B(t)$ be Brownian motion with $B(0)=x>0$ and let $A>x$. It is well known that the expected time for $B(t)$ to escape the interval $[0,A]$ is equal to $x(A-x)$.
Is the expected time known ...

**6**

votes

**2**answers

494 views

### Probability of Brownian motion to have a zero in an interval

I have what should be a very simple questions for Brownian motion experts...
Let $[a,b]$ be a given time interval. Let $f(x)$ be the probability that a linear Brownian motion with initial value $x$ at ...

**3**

votes

**1**answer

179 views

### Characterizations of the GOE/GUE family of distributions

This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural ...

**1**

vote

**1**answer

76 views

### The probability of Levy process staying at a point

Assume $X_{t}$ is a 1-dimensional Levy process on a probability $(\Omega, \mathcal{F}, P)$. For a fixed point $x$ in the state space and fixed $t\neq 0$, what's the value of $ P(\omega: ...

**5**

votes

**1**answer

128 views

### What are all the stationary and pointwise independent random processes?

In the 60's, I. Gel'fand introduced the concept of generalized stochastic processes (Ch. III, Vol. 4 of his work on Generalized functions). For a generalized stochastic process $\Phi$, he defines the ...

**7**

votes

**2**answers

380 views

### Can every discrete martingale be embedded in a continuous martingale?

Let $(X_k)_{k=0,1,..., n}$ be a discrete martingale defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to know whether there exists a (continuous) martingale ...

**3**

votes

**2**answers

125 views

### Initial paper of Gel'fand on Generalized Random Processes

The theory of generalized stochastic processes was introduced independently in the 50's by Ito* and Gel'fand in a short paper. The latter then developed his theory more extensively in the fourth tome ...

**2**

votes

**0**answers

128 views

### right-continuity of filtration

For a natural filtration of a stochastic process (possibaly multi-dimensional) to be right-continuous, what conditions should the process satisfy? Any references?

**1**

vote

**0**answers

164 views

### asymptotic variance of sample autocorrelation of two iid random variables

I am trying to prove that the variance of the sample lag-1 autocorrelation
$$\hat{\rho}=\frac{\sum_{t=1}^n(x_t-\bar{x})(x_{t-1}-\bar{x})}{\sum_{t=1}^n(x_{t-1}-\bar{x})^2}$$
for an i.i.d. R.V is ...

**1**

vote

**1**answer

75 views

### The jump and the left martingale of semimartingale

Let $X_{t}$ be a semimartingale. Define
$\Delta X_{t} = X_{t}- X_{t-}$.
For fixed $s> 0$, $\Delta X_{s}$ and $X_{s-}$ are two random variable. Are they independent to each other? I think the ...

**3**

votes

**1**answer

125 views

### The regularity of Levy process

There is a property for continuous Markov process that each point $y$ in its state space is hit with positive probability one starting from any interior point $x$.
This property is called the ...

**2**

votes

**0**answers

79 views

### Null sets visited infinitely often by trajectories of the shift dynamical system

Let $(G,\circ)$ be a Polish group, with identity $e$.
Let $\Omega$ be the set of continuous functions $\omega:\mathbb{R} \to G$ such that $\omega(0)=e$.
For each $t \in \mathbb{R}$, define the ...

**3**

votes

**0**answers

263 views

### Tight lower bound for expected maximum of K sums of T Rademacher random variables

For each $j \in \{1, \ldots, K\}$, let $(\varepsilon_{j,t})_{t=1}^T$ be an independent sequence of iid Rademacher random variables (i.e. taking values $\pm 1$ with equal probability). What is the best ...

**3**

votes

**0**answers

96 views

### The distribution of Jump gaps of Levy process

Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps ...

**0**

votes

**0**answers

275 views

### Sufficient condition for local martingale property of stochastic integral

Is the following correct and/or a (simple) known result?
Let $X$ be a local martingale and $H$ an integrand for $X$, such that the stochastic integral $\int H\cdot dX\ge x$ for some random variable. ...

**3**

votes

**1**answer

129 views

### Can this two-dimensional process self intersect?

I would like to know more about the two-dimensional processes derived from Brownian motion by the following stochastic differential equation (in the Ito sense)
$$dX_t = f(X_t) dt + ...

**5**

votes

**0**answers

308 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 ...