Sojourn time in an union of intervals for diffusions

Sojourns of Stationary Gaussian Processes over a Random Interval

We investigate asymptotics of the tail distribution of sojourn time $$ \int_0^T \mathbb{I}(X(t)> u)dt, $$ as $u\to\infty$, where $X$ is a centered stationary Gaussian process and $T$ is an independent of $X$ nonnegative random variable. The heaviness of the tail distribution of $T$ impacts the form of the asymptotics, leading to four scenarios: the case of integrable $T$, the case of regularly varying $T$ with index $λ=1$ and index $λ\in(0,1)$ and the case of slowly varying tail distribution of $T$. The derived findings are illustrated by the analysis of the class of fractional Ornstein-Uhlenb…

Joint distribution of the process and its sojourn time for pseudo-processes governed by high-order heat equation

Consider the high-order heat-type equation $\partial u/\partial t=\pm \partial^N u/\partial x^N$ for an integer $N>2$ and introduce the related Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study the sojourn time $T(t)$ in the interval $[0,+\infty)$ up to a fixed time $t$ for this pseudo-process. We provide explicit expressions for the joint distribution of the couple $(T(t),X(t))$.

局部自相似高斯过程的逗留时间

Sojourns of locally self-similar Gaussian processes

本文针对局部自相似高斯过程，推导了有限时间区间上累积巴黎破产概率的精确渐近近似，并证明Berman常数可由不同于分数布朗运动的自相似过程定义。

On the sojourn time of a Generalized Brownian meander

In this paper we study the sojourn time on the positive half-line up to time $ t $ of a drifted Brownian motion with starting point $ u $ and subject to the condition that $ \min_{ 0\leq z \leq l} B(z)> v $, with $ u > v $. This process is a drifted Brownian meander up to time $ l $ and then evolves as a free Brownian motion. We also consider the sojourn time of a bridge-type process, where we add the additional condition to return to the initial level at the end of the time interval. We analyze the weak limit of the occupation functional as $ u \downarrow v $. We obtain explicit distributiona…

A survey on the pseudo-process driven by the high-order heat-type equation $\partial/\partial t=\pm\partial^N/\partial x^N$ concerning the first hitting times and sojourn times

Fix an integer n>2 and let $(X(t))_{t\ge 0}$ be the pseudo-process driven by the high-order heat-type equation $\partial/\partial t=\pm\partial^N/\partial x^N$. The denomination "pseudo-process" means that $(X(t))_{t\ge 0}$ is related to a signed measure (which is not a probability measure) with total mass equal to 1. In this note, we present some results and discuss some problems concerning the pseudo-distributions of the first overshooting times of a single barrier $\{a\}$ or a double barrier $\{a,b\}$ by $(X(t))_{t\ge 0}$, as well as those of the sojourn times of $(X(t))_{t\ge 0}$ in the in…

Exponential moments of first passage times and related quantities for Lévy processes

For a Lévy process on the real line, we provide complete criteria for the finiteness of exponential moments of the first passage time into the interval $(r,\infty)$, the sojourn time in the interval $(-\infty,r]$, and the last exit time from $(-\infty,r]$. Moreover, whenever these quantities are finite, we derive their respective asymptotic behavior as $r \to \infty$.