# Questions tagged [stochastic-processes]

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

1,566
questions

**0**

votes

**0**answers

30 views

### Correlation of stopping times for integral of Brownian motion increment

Let $\mu(x):=\int_{\epsilon}^{x}\exp\{B_{s+\epsilon}-B_{s-\epsilon}\}ds$, where $(B_{s})_{s\geq 0}$ is a Brownian motion (starting at $B_{0}=0$) and epsilon is small $0<\epsilon\ll 1 $. Consider ...

**0**

votes

**0**answers

54 views

### Do Lyapunov functions imply exponential integrability of hitting times?

I have a question of some integrability of hitting times.
Let $X=(\{X_t\}_{t \ge0},\{P_x\}_{x \in E})$ be a diffusion process on a locally compact separable metric space $E$.
We assume that there ...

**2**

votes

**0**answers

49 views

### 2-d geometric Brownian motion hitting time distribution

I am trying to solve following problem: Given two independent geometric Brownian motions
$\frac{d x_t}{x_t}=\mu_x dt + \sigma_x dw_t^x$
and
$\frac{d y_t}{y_t}=\mu_y dt + \sigma_y dW_t^y$
and ...

**-1**

votes

**0**answers

24 views

### Dominance convergence theorem to compute expectation of a sequence of random variables defined by their time derivatives

Let $ (X_t) $ be a stochastic process, and define a new stochastic process by $ Y_t = \int_0^t f(X_s) ds $. Then consider a sequence $X_t^0,X_t^1,\ldots, X_t^n$ for which we get $Y_t^0,Y_t^1,\ldots, ...

**-4**

votes

**0**answers

24 views

### . Let B be a Brownian motion. Compute the mean of the random variable [closed]

$$
\xi=e^{2 B_{T}} \int_{0}^{T} e^{2 B_{t}+t} d B_{t}
$$

**-4**

votes

**0**answers

32 views

### Question on probability and random process [closed]

How can one define the variance of N-dimensional random variable ( X1, X2,...., XN ).Also how to compute the variance of a linear combination of N-random variables.

**-2**

votes

**1**answer

35 views

### Ensemble averaging in a random graph (or network) in the large $N$ limit [closed]

I have a random graph/network described by the adjacency matrix $(a_{ij})_{N\times N}$ where $a_{ij}=1$ with probability $p$. Each node in the graph is associated with a continuous quantity $\eta_i=\...

**1**

vote

**1**answer

82 views

### Eigenspace of Gaussian Markov operator

Consider the (one-dimensional) Gaussian distribution $Q := N(\nu,\tau^2)$ and the (Gaussian) Markov operator
\begin{equation*}
\begin{array}{rccc}
R : & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) &...

**2**

votes

**2**answers

115 views

### is this process a Markov one?

Here is the problem I can't solve.
Let $\xi_n$ $(n=1,2,3,\dots)$ be a sequence of i.i.d. random variables on $\mathbb{R}$ with density $p(x)>0$, let $\eta_n=\sum_{i=1}^{n}\xi_i^2$. Define $$\...

**0**

votes

**1**answer

80 views

### Finding a connection between two types of convergence

Please, help me find connections between two types of convergence:
Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences:
1) $X_n \...

**4**

votes

**2**answers

104 views

### Poisson counting process subinterval distribution

Suppose $N(\omega,t)$ is a homogeneous Poisson counting process with a constant parameter $\lambda,\,\forall\omega \in\Omega$ where $\Omega$ is the sample space. Given positive real numbers $T$ and $\...

**-1**

votes

**0**answers

72 views

### Weak convergence and Lipschitz function

I want to construct such r.v. ${ξ_n}$, $n≥1$$: (Ω,F,P)→(R^1,Bor)$, ${ξ_n}$ weakly converges to ${ξ}$ and such $f$ - Lipschitz function , so that $E(|f(ξ_n)−f(ξ)|) \not\to0$.
I tried to apply ...

**1**

vote

**2**answers

166 views

### Exponential or sub-exponential ergodicity?

Consider the one-dimensional stochastic differential equation $$d X(t) = -sgn(X(t))dt + dW(t),$$ where $W$ is a standard Brownian motion, and $sgn(x) = 1$ if $x > 0$ and $-1$ if $x\le 0$. It can be ...

**0**

votes

**2**answers

137 views

### Is there a generalised version of the Donsker invariance principle for a “sort-of continuous-time-random-walk”?

(The following question arises from my Math.SE question https://math.stackexchange.com/questions/3643865).
Let $\rho$ be a probability measure on $\mathbb{R} \times (0,\infty)$, and writing $\ \pi_1 \...

**1**

vote

**1**answer

54 views

### Kolmogorov tightness criterion for stochastic processes

I am searching for the criterion stated above and also here: The question about Kolmogorov tightness criterion.
It should state the following: If a sequence of stochastic processes $(X^n)$ fulfills:
...

**0**

votes

**1**answer

26 views

### Can the joint law $P \circ (X,Y)^{-1}$ of two random variables $X$ and $Y$ be written as $P \circ (X,\phi(X,U))^{-1}$ for $U$ uniform in $[0,1]$?

I want to know whether there is some general assumpitons we can make on two measurable spaces $E$ and $F$ (e.g. polish, complete, separable,...) such that we can ensure that the following "Theorem" ...

**-1**

votes

**0**answers

30 views

### Stopping times about Brownian motion with draft

Assumet $M(t) = B(t) + \mu t$ where $B(t)$ is a standard Brownian Motion. Denote:
$$T_a := \inf \{ t \geq 0, \, M(t) = a\}, \quad T_b := \inf \{ t \geq 0, \, M(t) = b\}$$
The question asks to ...

**1**

vote

**1**answer

62 views

### The weak convergence of finite dimensional distribution of Gaussian process does not imply the weak convergence in $C[0,1]$

In the study of weak convergence in $C[0,1]$, a common example is always being considered: $$X_{n}(t)=nt1_{[0,1/n]}(t)+(2-nt)1_{(1/n,2/n]}(t).$$ This example serves a counter-example to show that the ...

**0**

votes

**1**answer

58 views

### Find a conditional expectation of a difference of two independent Poisson process

Consider two independent Poisson processes $N,M$ with rate $\lambda$, and define $$X(t):=x+\dfrac{1}{\sqrt{n}}[N(t)-M(t)].$$ From this formula we know that $X(0)=x$. Now, I want to compute the ...

**0**

votes

**0**answers

14 views

### Calculate regime-switching correlation matrix without assumption on distribution

There's rich literature on Markov regime-switching dynamic correlation matrix. But most results seem to assume a certain kind of distribution and use MLE/EM. For example, some sort of multivariate ...

**2**

votes

**1**answer

127 views

### Average number of elements of a subset S of a matrix A after inducing the rows and columns of m randomly selected elements from subset S

Let $A_{N{\times}N}$ be an $N{\times}N$ matrix and $\mathcal{S_{k}}$ be a subset of elements in $A$ such that exactly $k$ elements from every row and column in $A$ are in $\mathcal{S_{k}}$. Thus, $\...

**2**

votes

**1**answer

81 views

### Proof of Hitting-time theorem in branching processes

I want to understand theorem 5.21 (page 224) in this link and here is where I don't understand: $$
\{W = t\} = \{t \text{ is the first ladder index in }R_1, \dots, R_t\},$$
i.e. $\{R_t = 1, R_1 < 1,...

**2**

votes

**1**answer

118 views

### Orthonormal frame bundles on a manifold

Let $(\mathcal{M},g)$ be a torsion free compact Riemannian manifold of dimension $n$. Hence from the metric we know there is an associated horizontal sub-bundle $H_u F \mathcal{M}$ of the orthonormal ...

**2**

votes

**2**answers

203 views

### Why the Komlós theorem is not valid for any sequence of measurable functions?

I read an article, and they use a certain theorem, called Komlós theorem, which says:
Theorem 1 (Komlós theorem)
Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $ (f_n)_{n\geq 1} \...

**0**

votes

**0**answers

35 views

### Existence of two stochastic processes

I am wondering if I can show that
For given $x,y\in \mathbb{R}$ there are two stochastic processes $S_t$ and $B_t$ such that $S_t$ and $B_t$ are two one dimensional Brownian motions starting at $x$ ...

**0**

votes

**0**answers

60 views

### Probability that random points are affine subspace

I asked this question in Math stack exchange, but I think it is more relevant here.
Suppose that $\mathbb{F}_q$ is a finite field with $q$ elements. Let $U = \{u_1, \ldots,u_m\}$ be a set of $m$ ...

**1**

vote

**1**answer

186 views

### General upper bound of extinction probability

We consider here a Galton–Watson process with an offspring distribution $X$, where $\mathbb{E}X = \mu$ and $\operatorname{Var} X = \sigma^{2} < \infty$ and $q = \mathbb{P}(\text{extinction})$, i.e.,...

**0**

votes

**0**answers

16 views

### Bifractional Brownian motion admit a representation in the form of a stochastic integral?

good morning. You know the fractional Brownian motion, multifractional Brownian motion and sub-fractional Brownian motion, can be represented as a wiener integral ( moving average representation ). ...

**0**

votes

**2**answers

60 views

### Show that if $A_{0}(t)+A_{1}(t)W(t)=0$ for all $t$ with $A_{0}$ and $A_{1}$ differentiable in $t$ and $W(t)$ a Wiener process, then $A_{0}=A_{1}=0$

I am learning the quadratic variation of stochastic process, and I am working on an exercise stating that
If for all $t$, we have $$0=A_{0}(t)+A_{1}(t)W(t),$$ where $(A_{0}(t),\mathcal{F}_{t})$ ...

**0**

votes

**0**answers

35 views

### When does the solution to the Fokker-Planck equation admit a density wrt Lebesgue measure?

Given a Markov process $(X_t)_{t\geq 0}$ on $(\mathbb R^n, \mathcal B_{\mathbb R^n})$, under which conditions does the solution to the Fokker-Planck equation
$$\frac{\partial u(t,x)}{\partial t}= \...

**0**

votes

**0**answers

22 views

### Sufficient condition for weak existence of solution of a SDE

Please be adviced that I'm cross-posting this question from MSE since it's very likely it will remain unsolved, and I haven't been able to obtain an answer from my colleges/professors.
It's a well ...

**2**

votes

**2**answers

137 views

### Process with covariance $E[Y_{t}Y_{s}]=a_{1}-a_{2}|t-s|$

We have a centered Gaussian process $X_{t}$ where we have exact equality $$E[X_{t}X_{s}]=a_{1}-a_{2}|t-s|$$ for $|t-s|<\epsilon_{0}\ll \frac{a_{1}}{a_{2}}$ and $a_{i}>0$.
Q: I am curious if ...

**2**

votes

**1**answer

68 views

### Proof of extended supermartingale convergence theorem

There is a supermartingale convergence theorem which is often cited in texts which use Stochastic Approximation Theory and Reinforcement Learning, in particular the famous book "Neuro-dynamic ...

**1**

vote

**0**answers

206 views

### On the level of measure theory, what does it mean for a drift to be deterministic?

Given a drift $F\in W^{1,2}([0,T])$ adapted to the filtration of a Brownian motion $B(t)$ on Wiener space $(C[0,T],\mathcal B(\|\cdot \|_\infty)$ with Wiener measure $\mu_0$, there is another measure $...

**1**

vote

**0**answers

17 views

### SDE conditional expectation

Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\
X_0=x_0 \\
dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...

**2**

votes

**0**answers

53 views

### Showing an “obviously-optimal” control is optimal (without smoothness assumptions)

Let $\mathcal{A}\subseteq\mathbb R$ be a compact interval, $T\in\mathbb R_+$ be a finite horizon, and $g:\mathbb R\to\mathbb R_+$ be a continuous function with $g\leq 1+|\cdot|$. Consider an optimal ...

**1**

vote

**1**answer

93 views

### Absolute value of a diffusion

Suppose $B_t$ is a standard Brownian motion on a filtered probability space $\langle \Omega, \mathcal F, \{\mathcal F_t\}_t, \mathbb P\rangle$. Consider two SDEs below.
Suppose, $X_0 = Y_0 = 0$
\...

**0**

votes

**0**answers

29 views

### Existence of an optimal control

I am looking for an existence result for the following control problem:
Fix a probability space $\langle \Omega, \mathcal F, \{\mathcal F_t\}_t, \mathbb{P}\rangle$ that satisfies the usual ...

**0**

votes

**1**answer

66 views

### Question about the exit time of a time-homogeneous Itô diffusion

Consider a one-dimensional Itô diffusion:
$$\mathrm{d} X_{t}=b\left(X_{t}\right) \mathrm{d} t+\sigma\left(X_{t}\right) \mathrm{d} B_{t}$$
where $X_0 = 0$ and $B_t$ is the standard Brownian Motion. ...

**0**

votes

**0**answers

8 views

### About the role of total variation measure on boundary reflected stochastic processes

I am reading this paper about stochastic differential equations with reflecting boundary conditions. In page 165, an example equation with an explicit solution is presented. However, I can't see that ...

**1**

vote

**2**answers

51 views

### Counterexample for absolute summability of autocovariances of strictly stationary strongly mixing sequence

Suppose $(X_i)_{i\in\mathbb{Z}}$ is a strictly stationary, strongly (i.e. $\alpha-$)mixing sequence of real random variables. If we have $\mathbb{E}[|X_1|^{2+\epsilon}]<\infty$ for some $\epsilon&...

**1**

vote

**0**answers

47 views

### Local time as a measurable map from Wiener space

Let $B$ be a Brownian motion on $[0,1]$. The local time of $B$, which I will denote by $L$, is defined as the process on $\mathbb R$ such that
$$\int_0^1 F(B_t)~dt=\int_\mathbb R F(x)L(x)~dx,\qquad\...

**0**

votes

**0**answers

70 views

### Ranking graph's nodes by score propagation

Problem
I have the following directed tripartite graph $G(E\cup V\cup P, A)$, where there is a many-to-one symmetric relationship between the subsets V and E - $e\in E,v\in V,[e, v]\in A \iff [v, e]\...

**1**

vote

**0**answers

44 views

### conditional expected value and in Stochastic differential equations

Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\
X_0=x_0 \\
dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...

**3**

votes

**0**answers

99 views

### Markov semigroups and resolvents, difference of continuity

Let $(E,d)$ be a locally compact separable metric space. We have a Markov process $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in E})$ on $E$. For bounded measurable function $f$ on $E$, we define
\begin{align*}...

**1**

vote

**0**answers

53 views

### Spitzer's condition, a slowly varying function and its behavior

Let $S$ denote a random walk that satisfies Spitzer's condition $$ \frac{1}{n} \sum _{k=1}^n P (S_k > 0 ) \to \rho$$ for some $\rho \in (0,1)$. From the book Regular Variation (Bingham, Goldie, ...

**2**

votes

**0**answers

64 views

### Tightness of Hilbert-space-valued arrays

Let $\mathcal{H}$ be a separable Hilbert space. Assume we have some triangular array $W_{n,j}, j=1, \ldots ,n $ of $\mathcal{H}$-valued random elements with $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}...

**0**

votes

**2**answers

61 views

### Martingale optional stopping before a stopping time

Here’s an easy one, I hope:
Suppose $\tau$ is a stopping time and $(M_t)$ is a martingale which together satisfy the hypotheses of the optional stopping theorem so that $\mathbb{E}[M_\tau]= \mathbb{E}...

**0**

votes

**1**answer

80 views

### Law of large numbers for Harris recurrent Markov chains

I'm trying to familiarize myself with the details of the proof that the Markov chains produce by the Metropolis-Hastings algorithm have a law of large numbers. I've found a half dozen or more ...

**0**

votes

**1**answer

58 views

### Stochastic invariant subset

Let us consider a stochastic differential equation (SDE),
$$
dx_{t}=f\left( x_{t}\right) dt+\sigma\left( x_{t}\right) dW_{t}%
$$
and a compact set $C\subset\mathbb{R}^{n}$.
Given a stochastic ...