Questions tagged [stochastic-processes]
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
2,354
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the infimum of a random process
Let $X_{t}=\sum_{i=1}^n(1+s\cdot w)\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w\sim\mathbb{N}(0,1)$, $s$ is a scalar denoting the strength of Gaussian noise. How to find the condition on $...
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Bounding random process
Def
$\{X_t\}_{t\in T}$ is called Lipschitz for metric $d$ on $T$ if there exists a random variable $C$ such that
$$|X_t-X_s|\leq Cd(t,s),\text{ for all }t,s\in T.$$
Lemma
Suppose $\{X_t\}_{t\in T}$ is ...
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Can independent Brownian motions hit zero at the same time?
Consider for $i=1,\ldots, N\ge2$
$$X^i_t=x_i+W^i_t,\quad \forall t\ge 0,$$
where $x_1,\ldots, x_N\in (0,\infty)$ and $W^1,\ldots, W^N$ are independent Brownian motions. Denote by $\tau_i$ the first ...
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Ergodicity of the solution to some SDE
Consider the SDE (stochastic differential equation) as follows:
$$dX_t=X_t\big(b(X_t)dt+a(X_t)dW_t\big)$$
where $b,a:\mathbb R\to\mathbb R$ are Lipschitz and bounded and $W$ is a real-valued Brownian ...
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Bounding from below the distance between SDE started from different initial conditions
Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t$$
with $\mu, \sigma: \mathbb R \to \mathbb R$ Lipschitz ...
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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^-&...
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Recurrence of Drifted Brownian Motion Conditioned to not hit Moving Barrier
Suppose we have a Brownian motion $X$ with $X_0>0$ and drift $\mu$ conditioned to be less than a barrier $R$ which has behaviour $R_0 = r$, $dR_s = \nu \, ds$, where $\mu > \nu > 0$.
Can we ...
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Lower bound of $\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]$ (without observing history)
Let $X$ be the solution to some stochastic differential equation
$$dX_t =b(X_t) \, dt+a(X_t) \, dW_t,\quad \forall t>0.$$
Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ ...
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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 ...
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Stationary Distribution of Langevin Dynamics driven by Lévy Process
Let $f\geq 0$ be a Lipschitz function and let $(L_t)_{t\geq 0}$ be an $\alpha$-stable Lévy process ($0<\alpha<2$, possibly multivariate). Consider the process given by $$dX_t=-\nabla f(X_t)dt+\...
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How can we prove that a stochastic process converges to a deterministic value?
As an illustrative example, consider a modified O-U process $dX_t = -X_tdt + \exp(-t)dW_t$. It is not too hard to understand that after a while the behaviour is dominated by the deterministic ...
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Version of Kolmogorov tightness criterion without moments
Kolmogorov tightness criterion says that if $X_N$ is a sequence of continuous process with $X_N(0)=0$ and $E[[X_N(t)-X_N(s)|^p]\leq C_p |t-s|^{1+\beta}$ then for all $\gamma\in (0,\beta/p)$ we have ...
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Hölder continuity of process from Donsker like theorem with Cauchy random variables
Let $X_k$ be i.i.d. Cauchy random variables with parameters $0,1$. For each $N$ define the process $Y_N$ by $$Y_N(t)=\frac{1}N\sum_{k=1}^{\lfloor tN\rfloor}X_k+\text{piecewise linear interpolation}.$$
...
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Asymptotic behaviour of the solution to some delayed stochastic differential equation
Consider the delayed stochastic differential equation as below:
$$dX_t^\theta=X_{(t-\theta)^+}^\theta(1-X_{(t-\theta)^+}^\theta)(dt+dW_t),\quad \forall t>0$$
$$dY_t^\theta=Y_{(t-\theta)^+}^\theta(1-...
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Characteristic function of stochastic integral of a pure jump Lévy process with respect to another pure jump Lévy process
(I am cross-posting this question here from MSE: https://math.stackexchange.com/questions/4725734/characteristic-function-of-stochastic-integral-of-a-pure-jump-l%c3%a9vy-process-with. I apologize if ...
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Convergence of stochastic linear recurrences
Suppose that $\zeta_t$ is a univariate, stationary stochastic process ($t\in\mathbb{N}^+$).
Let $x_0\in\mathbb{R}^n$, and let $f:\mathbb{R}\rightarrow\mathbb{R}^{n\times n}$ be a continuously ...
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Can a diffusion process admit an invariant measure with a non-differentiable density?
The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...
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Escaping probability of a Brownian particle in random enviroment
Let $\Omega\subset \mathbb R^d$ be a bounded open (and connected) set. Consider $E\subset \Omega$ and $x\in \Omega\setminus E$. Denote by $W^x$ the standard Brownian motion starting at $x$, i.e. $W^...
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A question about convergence of stochastic processes converging to a random walk
Consider the following random walk $(y_t)_{t \in \mathbb Z_+}$:
$$y_t = y_{t-1} + u_t,\quad (u_t)_{t \in \mathbb Z_+} \overset{iid}{\sim} N(0,1), \quad (t \in \mathbb Z_+)$$
where $y_0, u_1, u_2,...$ ...
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Garsia-Rodemich-Rumsey without Markov
Let $X$ be a $\mathbb R^d$ valued continuous stochastic process. I am interested in bounding $$P(\|X\|_\gamma>R).$$
The standard technique to do so, is to apply Markov inequality and then Garsia-...
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SPDEs driven by fractional brownian noise
I am looking for some references for the following kind of SPDEs
$$dX_t= AX_t\,\mathrm{d}t+BX_t\,\mathrm{d}W^H_t,$$
given $X(0)=X_0$, where $A$ and $B$ are operators and $W^H_t$ is the fractional ...
3
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Request for references of random matrices
I need some good books aimed as a detailed and gentle introduction to random matrices, containing good discussion and derivation of Marchenko–Pastur distribution. Also, I request some other references ...
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Functional integral formulas for the wave equation and other hyperbolic PDEs
The Feynman–Kac formula provides a functional (Wiener) integral representation of the solution $u$ to the heat equation
\begin{align*}
\partial_t u &= \frac{1}{2}\Delta_x u,\\
u(0,x) &= ...
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Martingale regularization
Consider a submartingale $X,$ then for almost every $\omega \in \Omega,$ for every $v \in \mathbb{R},\lim_{u \in \mathbb{{Q},u \uparrow v}}X_u(\omega)$ exist in $\mathbb{R}.$
I was wondering if there ...
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Characteristic function of quadratic variation of compound Poisson process
If I have a compound Poisson process whose characteristic function is known, is there a way to calculate the joint characteristic function of this process and its quadratic variation process?
If not ...
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137
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joint density of two relevant random variables
It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...
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"Ergodic theorem" for Markov kernels
Consider a discrete time Markov chain $(X_t)$ on a finite state space $\mathcal{S}$, with transition matrix $P$. Assume that the chain admits a stationary distribution $\pi$, which I will identify ...
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Step in the derivation of the total idle time distribution of an M/G/1 queue
I'm trying to work my way through the proof of Thm. 1.11 in Kyprianou's Introductory Lectures on Fluctuations of Levy Processes with Applications but really struggle to understand the following step. ...
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Maximise the probability that a drifted Brownian motion doesn't hit zero prior to $T$
Let $W=(W_t)_{t\ge 0}$ be a standard Brownian motion starting from zero and $Z>0$ be an independent random variable. Fix $T>0$ and $C>0$. Denote by $\mathcal A$ the set of progressively ...
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Characteristic function of a Dirichlet Process
Suppose $P \sim \text{DP}(\alpha,G) $ where $G \sim N(0,1)$ is the base measure and $\alpha > 0$ is the concentration parameter. The stick breaking representation says that $P$ can be expressed as \...
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Randomly chosen walk of fixed length
Let $G=(V, E)$ be the graph on vertices $V = \{0, \cdots, k\}^n$, where vertices $(v_1, \cdots, v_n)$ and $(w_1, \cdots, w_n)$ share an edge iff $\lvert v_i - w_i\rvert \leq 1$ for all $i$.
A walk of ...
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(Rate of) Convergence in distribution and Laplace transform of random variables/stochastic processes
Let $X_t^n$ and $X_t$ be stochastic processes (with finite moments), and assume that for every $t>0$, $\lambda>0$ and bounded continuous function $\varphi$,
\begin{equation}
\int_0^te^{-\lambda ...
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Markov chain to solve a particle fusion problem
A sequence of elementary particles arrive at Poisson rate $r$ to a system. A pair of elementary particles can be fused into a level-$1$ particle. The fusion process succeeds with probability $p_0$. ...
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How to find lower bounds of a modified mixing time (defined below) with respect to spectral of a finite Markov chain?
I am focused on a time-homogeneous continuous-time Markov chain with a finite state space $\mathcal{X}$, whose Markov kernel is $K$ and the corresponding semigroup is $H_t=e^{-t(I-K)}$. The invariant ...
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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, ...
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Asymptotic behavior of a Markov process on the set of $\{0,1\}$-polynomials
This question is cross-posted from https://math.stackexchange.com/questions/4711799/asymptotic-behavior-of-a-markov-process-on-the-set-of-0-1-polynomials
I am trying to study the asymptotic behavior ...
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Concatenation of Markov processes and independence
In chapter 14 of Sharpe's General Theory of Markov Processes the concatenation of Markov processes $X^1$ and $X^2$ is described. I've posed the relevant part at the bottom of this post.
It is rather ...
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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 $...
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Branching process with varying offspring distribution at each step
Consider a simple branching process $Z_0,Z_1,Z_2...$ such that at every discrete step, a particle splits into $k\geq1$ particles where $k$ follows a discrete distribution with probability mass $p(k)$.
...
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Variance of the logarithm of the mixed Rademacher and complex Gaussian distribution
Consider the scenario where $X$ is a Rademacher random variable taking values $\{−1,+1\}$ with equal probability, and $Z$ is a complex Gaussian random variable with a mean of $0$ and a variance of $\...
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Can we use epsilon-net method on an open set?
Let $X_{t\in T}$ be a random function where $T$ is a subset of $\mathbb{R}^n$.
Since $T$ has inifite points, we are not able to use union bound to estimate $\sup X_t$. Thus instead, when $T$ is ...
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Strong blow up limits for SDE
Note: This is a strengthening of the following result, motivated by the need for strong convergence in applications.
Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution ...
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Eigenvalues/eigenfunctions of a diffusion generator
Consider the following symmetric second order diffusion operator, defined, for $\phi \in \mathcal{C}^{2,1}_c\left(\mathbb{R}\times \mathbb{R}_+\right)$, by:
$$L\phi := \lambda_1 \partial_{R_1}(R_1 \...
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Regularity of a function depending on first exit time of martingale
Consider a parametrised martingale as follows :
$$X^x_t := x+ \int_0^t\sqrt{2p_s} \, dW_s,$$
where $W$ is a standard Brownian motion and $(p_t)_{t\ge 0}$ is a locally square integrable process ...
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Does the entropy of a SDE with nondegenerate noise always increase?
Let $W$ be a standard Brownian motion, and let $X$ be the solution to the one dimensional SDE
$$dX_t = \sigma(t, X_t) \, dW_t$$
with initial condition $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. We ...
2
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Convergence of the quadratic variation process
Suppose we are given a sequence of stochastic processes $X^n, n\in\mathbb{N},$ with finite quadratic variations and a stochastic process $X$ such that for every $t\geq0$
$$
\lim_{n\to\infty}\mathbb{E}(...
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Local martingale for a (two-dimensional) diffusion
Let $X$ be a two-dimensional diffusion (a solution of $dX_t=f(X_t)\,dt+dB_t$, with $B$ a standard two-dimensional Brownian motion) living on some open set $\Lambda\subset \mathbb{R}^2$. Let $h:\Lambda ...
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73
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Calculation of the difference of two Brownian bridges
I was told that the difference of two independent brownian bridge process is $\sqrt{2}$ times a brownian bridge process, i.e.,
$$B_{1t} - B_{2t} = \sqrt{2}B_t$$
where $B_{1t}$ and $B_{2t}$ are ...
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A notion of SDE via the martingale representation theorem
$\newcommand{\d}{\mathrm{d}}$It is well-known that differentiating stochastic processes with respect to time is usually impossible in the usual sense. For instance, a Brownian motion $W$ on a ...
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Probability distribution for a Bayesian Update
I am struggling with a process like this:
$$X_t=\begin{cases}
\frac{\alpha\omega_t}{\alpha\omega_t+\beta(1-\omega_t)} & \text{with prob } p\\
\frac{(1-\alpha)\omega_t}{(1-\alpha)\omega_t+(1-\beta)(...