Questions tagged [stochastic-processes]

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

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

$$ \xi=e^{2 B_{T}} \int_{0}^{T} e^{2 B_{t}+t} d B_{t} $$
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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.
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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=\...
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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) &...
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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 $$\...
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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 \...
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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 $\...
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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 ...
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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 ...
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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 \...
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1answer
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: ...
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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" ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, $\...
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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,...
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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 ...
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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} \...
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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 $x$ ...
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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 $m$ ...
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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.,...
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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 ). ...
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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 $(A_{0}(t),\mathcal{F}_{t})$ ...
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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}= \...
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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 ...
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2answers
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 ...
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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 ...
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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 $...
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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^...
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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 ...
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1answer
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, $X_0 = Y_0 = 0$ \...
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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 ...
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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. ...
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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 ...
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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&...
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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\...
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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]\...
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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^...
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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*}...
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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, ...
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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}...
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2answers
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}...
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1answer
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 ...
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1answer
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 ...

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