Questions tagged [stochastic-processes]
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
866
questions with no upvoted or accepted answers
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views
Tail for the integral of a diffusion process
I would like to compute the following tail,
$$
\mathbb{P}\left(\int_{0}^{T} f(X_t)\mathrm{dt}>x\right),
$$
assuming
$$
\mathbb{P}[f(X_t)>x] = x^{-\alpha} \log(x),
$$
and $X$ is a diffusion ...
4
votes
0
answers
150
views
A simplified MCMC / MH algorithm. Are there known convergence results?
Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...
4
votes
0
answers
115
views
Sufficiency of stationary policy for negative stochastic dynamic programming
Consider a Markov Decision Process with Borel state space $X$ and Borel action space $U$, like the one defined in the book "Stochastic Optimal Control: Discrete-time case" by Bertsekas and Shreve. All ...
4
votes
0
answers
267
views
Some constants in Martingale Stein inequality
Dear all,
the following is a special case of Stein inequalities for martingales.
$\textbf{Theorem}$ Let $(\Omega, \mathbb{P})$ be a (standard) probability space equipped with a filtration of ...
4
votes
0
answers
1k
views
The spectrum of a Markov Operator and Invariant Measures
Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...
4
votes
1
answer
787
views
A balls into bins problem with combinatorial constraints
We are given $m$ balls and $n$ bins, with $m \ge n$. Each bin can contain at most $c$ balls (we assume that $c$ is an even integer). In a sequential fashion, at each time step, one ball is placed into ...
3
votes
0
answers
73
views
A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal
I apologize if this is too elementary a question, but I have not been able to make much progress.
Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. ...
3
votes
0
answers
69
views
Norm estimate for parabolic SPDE solution
When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
3
votes
0
answers
78
views
Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set
Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. ...
3
votes
0
answers
88
views
Explicit example of drift $F$ so that the law of $F+B$ is not absolutely continuous with respect to $B$
Let $\mu_0$ be the law of Brownian motion on the space of continuous functions. If $\mu\sim\mu_0$ agrees on null sets then there is some progressively measurable $F\in W_0^{1,2}$ a.s. so that $\mu$ is ...
3
votes
0
answers
173
views
Flow property for semimartingale driven SDE at a stopping time
Let $S$ be an $n$-dimensional semimartingale such that the SDE
$$dX_t = \sigma(X_t, t) \, dS_t$$
with $\sigma$ Lipschitz continuous admits a globally defined unique strong solution on $[0, T]$.
For $t ...
3
votes
0
answers
121
views
Stochastic braids
I am definitely not a probability guy, but I'd like to have a heuristic answer to the following question: do $n$ independently moving points in an open, connected, bounded region $R$ tend to "...
3
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0
answers
51
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Multi-type Galton-Watson-like process where only majority-type is allowed to reproduce
Are you aware of any research papers that have explored a multi-type Galton-Watson process in which only particles of the majority type are permitted to reproduce in each generation?
I've been unable ...
3
votes
0
answers
64
views
Algebra core for generator of Dirichlet form
This is a question about the existence of a core $C$ for the generator $A$ of a regular Dirichlet form $\mathcal{E}$ having a carré du champ $\Gamma$, so that $C$ is an algebra with respect to ...
3
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0
answers
65
views
Inverse comparison principle for stochastic differential equations
Consider two SDEs (stochastic differential equations) as follows:
$$dX_t=b^-(t,X_t) \, dt+a(t,X_t) \, dW_t;\quad dY_t = b^+(t,Y_t)\,dt+a(t,Y_t)\,dW_t,$$
where $b^-,b^+,a$ are Lipschitz such that $b^-&...
3
votes
0
answers
133
views
Request for article in Rev. Roumaine Math. Pures Appl. (1981)
I am looking for the following article:
Al-Hussaini, A. N. A projective limit view of $L_1$-bounded martingales.
Rev. Roumaine Math. Pures Appl.26 (1981), no.1, 51–54, but I can't find it anywhere.
Do ...
3
votes
0
answers
164
views
Asymptotics for optimal survival time in a stochastic control problem
An individual, henceforth called the runner starts at the center of an open ball $\Omega_r \subset \mathbb R^2$ of radius $r > 1$.
At each turn, a vector $x \in S^1$ is chosen uniformly at random, ...
3
votes
0
answers
94
views
Dealing with noise that is white in time, colored in space numerically
I am broadly working on a dynamic process where we want to see how a field $\rho(r)$ changes in space in time with thermal noise. The system is biased around a thermodynamic saddle point dictated by $...
3
votes
0
answers
67
views
The boundary between infinite clusters connected by closed and open bonds
In the following, I'll heuristically describe a boundary between two infinite clusters arising in percolation on the triangular lattice. I expect this concept has been well-studied before. My hope is ...
3
votes
1
answer
402
views
Positive definiteness of a matrix-valued function
This question is a repost from math.se, where I didn't receive an answer.
Are there simple conditions on an $d \times d$ matrix B under which
$$
f(t, s)
=
\begin{cases}
\exp(-B |t - s|^\alpha), &...
3
votes
0
answers
177
views
Example of an optional non-predictable process
To clarify better the notions of predictable and optional processes, I am looking for a simple example of a process that is optional, but not predictable. I found out something useful here, however, I ...
3
votes
0
answers
170
views
Elworthy’s 1982 “Stochastic Differential Equations on Manifolds” - relevant?
In 1982, D. Elworthy published “Stochastic Differential Equations on Manifolds”. Apparently, this was quite a seminal book in the field of stochastic DE’s/processes on manifolds. Is this reference ...
3
votes
0
answers
189
views
Probabilistic optimization problem on tree vertex selection without replacement proportional to the degree
We are given a tree $T(V,E)$ with $|V|=n$ vertices, where $V=\{v_1,v_2,\ldots, v_n\}$. We denote by $d_i$ the degree of vertex $v_i$ for all $i\in\{1,2,\ldots,n\}$.
In a sequential fashion, we select ...
3
votes
0
answers
90
views
Martingale problem for regime switching SDE
Let $\mu$ be a continuous time Markov process switching between two possible values $a, b \in \mathbb R$ with time homogeneous transition rates. Consider the one dimensional SDE
$$dX_t = \mu \, dt + ...
3
votes
0
answers
278
views
Math videos featuring interesting data animations
I am looking for interesting videos featuring pure data animations (not someone talking about math, but a video featuring some math phenomenon). I am interested in videos that tell a story, rather ...
3
votes
0
answers
466
views
The distribution of collision stopping time in 2D random walk
Assume two particles A at $(0, 0)$ and B at $(a, b)$ in 2D discrete grid, both of them have the same possibility of $\frac{1}{4}$ for moving up/down/left/right (i.e. 2D random walk). We define the ...
3
votes
0
answers
49
views
Conditions of parameters to have bounded solution of Dynkin's equation in exit problem
Consider the following Dynkin’s equation in exit problem defined on unit disk $D_1(0)$
\begin{align}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma-1}{r} \sin\psi \frac{\partial}{\partial\...
3
votes
0
answers
36
views
On the property of a nonnegative stochastic process "attracted" near zero
Let $\{X_k\}$ be a nonnegative stochastic process satisfying
$$E\left[ X_{k+1} \mid \mathcal{F}_k \right] \leq \rho X_k + c,$$
where $0 < \rho < 1, c>0$. Intuitively, the process is likely to ...
3
votes
0
answers
185
views
Regularity of harmonic functions for a degenerate elliptic operator
This is a question on a degenerate elliptic operator.
Let $E$ be a closed unit ball in $\mathbb{R}^d$ centered at the origin. For a positive number $c>0$ and $f \in C^2(E)(:=C^2(\mathbb{R}^d)|_E)$, ...
3
votes
0
answers
92
views
Probability measure on $\mathbb{R}^n$ with given marginals and given correlation matrix
In all what follows, let $\mathcal{P}(\mathbb{R}^n)$ denote the set of probability measures on $(\mathbb{R}^n, \mathcal{B}(\mathbb{R}^n))$ and $\mathcal{C}_n$ the set of $n \times n$ correlation ...
3
votes
0
answers
106
views
Continuous time Markov chains and invariance principle
This question may be elementary for experts
Let $\{\xi_n\}_{n=1}^{\infty}$ be an i.i.d random variables on a probability space $(\Omega,\mathcal{F},P)$. We assume that the mean of $\xi_n$ is zero, and ...
3
votes
0
answers
232
views
Probability of a particle surviving forever
Consider a particle whose position is driven by the following equation:
$$Y_t = y + t + W_t + C\min\big(1,(Y_t+1)^+\big)\Lambda_t,\quad \mbox{for all } 0\le t<\tau_*,$$
where $y>0$, $0<C<1$...
3
votes
1
answer
249
views
Optimal rule for multiple stopping times for defect finding
Suppose a quality inspector is inspecting $b$ black amongst which $d_B$ are known to be defective and $w$ white gadgets amongst which $d_W$ are known to be defective. The gadgets come down along an ...
3
votes
0
answers
78
views
Random walk in a switching scenery
For each $x \in \mathbf{Z}$ let $(\eta_t(x))_{t\geq0}$ denote independent copies of a process $(\eta_t(0))_{t\geq0}$ defined as follows. The process $\eta_t(0)$ takes values in $\{-1,1\}$, where $-1$ ...
3
votes
0
answers
135
views
An integral involving Levy process with no positive jumps
Let $L_t$ be a Levy process that has no positive jumps, but is not strictly decreasing, i.e
$$
L_t = \gamma t + \sigma B_t + J_t,
$$
where $B_t$ is a Brownian motion, $J_t$ is a pure jump process with ...
3
votes
0
answers
117
views
The domains of generators of Markov processes and moment of hitting times
This is a question on domains of generators and moment of hitting times.
Let $M$ be a locally compact separable metric space and $X=(\{X_t\}_{t \ge 0}, \{P_x\}_{x \in M})$ a diffusion process on $M$ (...
3
votes
0
answers
81
views
How does one define the gradient of a Markov semigroup?
In the context of functional inequalities for Markov semigroups $(\mathcal P_t)_{t\ge0}$, what is one denoting by $\nabla\mathcal P_tf$? For example, I've found the following assumption in this paper:
...
3
votes
0
answers
135
views
Correlation of stopping times for integral of Brownian motion increment
Let $\mu(x):=\int_{\epsilon}^{x}\exp\{B_{s+\epsilon}-B_{s-\epsilon}\}ds$, where $(B_{s})_{s\geq 0}$ is a Brownian motion (starting at $B_{0}=0$) and epsilon is small $0<\epsilon\ll 1 $. Consider ...
3
votes
0
answers
147
views
Markov semigroups and resolvents, difference of continuity
Let $(E,d)$ be a locally compact separable metric space. We have a Markov process $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in E})$ on $E$. For bounded measurable function $f$ on $E$, we define
\begin{align*}
...
3
votes
0
answers
82
views
Have stick-breaking priors with non-iid atoms been considered, and if not, why not?
Roughly speaking, a stick-breaking prior is a random discrete probability measure $P$ on a measurable space $\mathcal X$ of the form
$$P=\sum_{j\ge1}w_j\delta_{\theta_j}$$
where $(w_j)_{j\ge1}$ is a ...
3
votes
0
answers
147
views
Beta distribution and Wiener process
Suppose $W(t)$ is the standard Wiener process on $[0; 1]$ and $\{T_x\}_{x \in \mathbb{R}}$ is a collection of random variables defined by the following relation:
$$T_x = \mu(\{t \in [0;1] | W(t) > ...
3
votes
0
answers
72
views
Random Two-Player Asymmetric Game
About half a year ago I asked a question on MSE about a random two player game. At the time, the question received some attention and some progress was made, but was not resolved completely. I have ...
3
votes
0
answers
173
views
Convergence rate of the smallest eigenvalue of an integral of a multivariate squared Brownian Motion
I am interested in deriving the convergence rate of the smallest eigenvalue of a sequence of random matrices with diverging dimension. More precisely, let $W_n(r)$ represent an $n$-dimensional ...
3
votes
0
answers
484
views
Domain of the Generator of a Bessel process
Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...
3
votes
0
answers
87
views
Mutual dependencies of BSDE solutions with markovian drivers with different starting points
Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$.
Let ...
3
votes
0
answers
321
views
Random walk on $\mathbb{R}$ with "sticky" origin
Let $P_i$, $N_i$, and $Z_i$, $i\in\mathbb{N}$ be r.v.'s with the $P_i$, $N_i$, and $Z_i$ being identically distributed with known pdf's $f_P$, $f_N$, and $f_Z$, respectively; and with no dependence ...
3
votes
0
answers
172
views
product of right continuous filtrations is right-continuous?
Let $\mathcal{G} = \sigma \lbrace G_{1},..., G_{n} \rbrace$ where $G_{1},..., G_{n}$ are subsets of $\Omega_{1}$ and $(\mathcal{F}_{t})$ is a right-continuous filtration on $\Omega_{2}$. Is $(\mathcal{...
3
votes
0
answers
448
views
Galton Watson tree with various kinds of offspring
As far as I understood, for the Galton-Watson tree process, the offspring are of one type. I am thinking of the case where we have offspring of different types. I have illustrated this in the example ...
3
votes
0
answers
77
views
Generators and Covariance Operators of Diffusions
For a constant coefficient Ornstein-Uhlenbeck process, how should I think about the relationship between the infinitesimal generator of the process and the covariance operator of the process (or, ...
3
votes
0
answers
84
views
Why is the Jain Monrad condition the right condition on general Gaussian processes?
Consider a covariance function $\sigma^2(s,t)=E((X_t-X_s)^2)$, where $X\colon I\to \Bbb R^d$ is a Gaussian process.
Given a $\rho\ge 1$ and a superadditive function $\omega(s,t)$ we say that Jain ...