# Questions tagged [stochastic-processes]

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

**4**

votes

**4**answers

2k views

### Time integrals of diffusion processes

I was wondering if someone could recommend a reference that deals with time integrals of diffusion processes.
Suppose $X$ is an Ito diffusion process with dynamics
$dX_t = \mu(X_t)dt + \sigma(X_t)...

**5**

votes

**1**answer

524 views

### Does generator of continuous time random walk map heat kernel from L^2 to L^2?

Let $\Gamma = (G,E)$ be an undirected, infinite, connected graph with no multiple edges or loops. We equip $\Gamma$ with a set of edge weights $\pi_{xy}$, where, given $e=\{x,y\}\in E$, we write $\...

**2**

votes

**2**answers

805 views

### Counterexample Markov process

Let $X$ be a homogeneous Markov process in a continuous time with value in the set $E$. Suppose that for some $T>0,x\in E, A\subset E$ we have
$$
P_x[X_t\in A] = 0
$$
for all $t\in [0,T]$ but
$$
...

**1**

vote

**1**answer

983 views

### Autocorrelation of a ±1-valued random process with certain statistics

Suppose $f(t)$ is a continuous-valued, zero-mean stochastic signal with Gaussian autocorrelation (with variance $\sigma^2$). Suppose I then pass this signal through a step function, producing a new $\...

**1**

vote

**2**answers

1k views

### Change of measure Markov process

We begin with example. For the Poisson process with an intensity $\lambda_1$ there is an equivalent change of measure which makes it intensity to $\lambda_2$.
I would like to find the conditions ...

**2**

votes

**2**answers

943 views

### Change of time or change of measure

Consider simple diffusion $dX_t = \sigma dw_t$ and a parameter $a>0$ and $X_0=x$. Let us denote $Y_t = X_{at}$ - thus we made a change of time. Let us denote an original measure as $P$. How to find ...

**0**

votes

**1**answer

937 views

### Generalizations of a product formula for the gamma function

Hello and Happy holidays.
I am interested in generalizations of the following product formula for the gamma function
$\Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$:
\begin{align}
\...

**2**

votes

**1**answer

530 views

### Reachability for Markov process

Let $X$ be a Markov process (in continuous or discrete time) and define an event
$$
R(T,A) = (\exists t\leq T: X_t \in A).
$$
I have seen in one paper that
$$
\Pr[R(\infty,A)] = \sup\limits_{\tau} \...

**8**

votes

**2**answers

569 views

### Random walks on graphs: Cover time and blanket time

Winkler and Zuckerman conjectured that the blanket time is within a constant factor of the cover time. The conjecture was recently proved. The cover time $C$ is the expectation of the first time $t$ ...

**4**

votes

**3**answers

582 views

### A type of stochastic jump process

Let $X \geq 1$ be a integer r.v. with $E[X]=\mu$. Let $X_i$ be a sequence of iid rvs with the distribution of $X$. On the integer line, we start at $0$, and want to know the expected position after we ...

**2**

votes

**1**answer

265 views

### A random variable in a game of knights and queens

Suppose that a game is played on an $n \times n$ board as follows. There are two players, Player 1 has (only) $Q$ queens and Player 2 has only $K$ knights. Suppose that $Q, K \leq n/3$. The game is ...

**0**

votes

**2**answers

2k views

### Two dimensional brownian motion first passage time

Hello,
I am looking for information on how to solve/compute first passage time for two dimensional Brownian motion.
any papers, references, books or web links for study will be helpful.
thanks
...

**5**

votes

**2**answers

3k views

### Commuting supremum and expectation

Given a one-parametric random function on a probability space $(\Omega,\mathcal F,\mathbb P)$:
$X:U\times\Omega\to \mathbb R \text{ and } (a,w)\mapsto X(a,w), \text{ with } \sigma(X(a))\subseteq \...

**1**

vote

**1**answer

857 views

### Discounted total reward vs. Average total reward

In a Markov Decision Process (MDP), the discounted total reward is defined as $\sum_{t=0}^\infty \gamma^tr_t$ where $r_t$ is the reward perceived at time $t$ and $\gamma$ is a real number $\in ]0, 1[$....

**3**

votes

**1**answer

516 views

### Stationary non-isotropic spatial stochastic processes

I asked this question in math.stackexchange but got no response;
Are there any interesting examples of second order stationary processes on ${\mathcal R}^2$ or ${\mathcal R}^3$ that are not isotropic?...