Nikolai Dokuchaev
First exit times from regions and their dependence on variations of boundaries are discussed for diffusion processes. The paper presents an estimate of $L_1$-distance between exit times from two regions via expectations of exit times.
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We introduce the exit time finite state projection (ETFSP) scheme, a truncation-based method that yields approximations to the exit distribution and occupation measure associated with the time of exit from a domain (i.e., the time of first passage to the complement of the domain) of time-homogeneous continuous-time Markov chains. We prove that: (i) the computed approximations bound the measures from below; (ii) the total variation distances between the approximations and the measures decrease monotonically as states are added to the truncation; and (iii) the scheme converges, in the sense that…
Efficient Importance Sampling for Wrong Exit Probabilities over Combinatorially Many Rare Regions
本文研究轻尾随机游走从众多罕见区域错误出口概率的重要性采样估计，提出通过结合少量区域的最优倾斜与控制方差的额外提议来构建分量更少的渐近高效混合方法，解决了高维下指数级区域导致的计算不可行问题。
We establish general moment estimates for the discrete and continuous exit times of a general Itô process in terms of the distance to the boundary. These estimates serve as intermediate steps to obtain strong convergence results for the approximation of a continuous exit time by a discrete counterpart, computed on a grid. In particular, we prove that the discrete exit time of the Euler scheme of a diffusion converges in the L1 norm with an order 1/2 with respect to the mesh size.
We study the sharp order of integrability of the exit position of Brownian motion from the planar domains ${\cal P}_α= \{(x,y)\in \bR\times \bR\colon x> 0, |y| < Ax^α\}$, $0<α<1$. Together with some simple good-$λ$ type arguments, this implies the order of integrability for the exit time of these domains; a result first proved for $α=1/2$ by Bañuelos, DeBlassie and Smits \cite{ba} and for general $α$ by Li \cite{li}. A sharp version of this result is also proved in higher dimensions.
We supplement a very recent paper of G. Markowsky concerned with the expected exit times of Brownian motion from planar domains. Besides the use of conformal mapping, we apply results from potential theory. We treat the case of a wedge-shaped region exactly, subsuming an example of Markowsky. A number of other results are presented, including for a half disc, $n$-grams, and a variety of other regions.
We investigate the tail distribution of the first exit time of Brownian motion with drift from a cone and find its exact asymptotics for a large class of cones. Our results show in particular that its exponential decreasing rate is a function of the distance between the drift and the cone, whereas the polynomial part in the asymptotics depends on the position of the drift with respect to the cone and its polar cone, and reflects the local geometry of the cone at the point where the drift is orthogonally projected.
It is given an effective upper estimate of expectation of |T_1-T_2|, where T_1 and T_2 are the first exit times from a region for two vector diffusion processes.
The purpose of this paper is to consider the exit-time problem for a finite-range Markov jump process, i.e, the distance the particle can jump is bounded independent of its location. Such jump diffusions are expedient models for anomalous transport exhibiting super-diffusion or nonstandard normal diffusion. We refer to the associated deterministic equation as a volume-constrained nonlocal diffusion equation. The volume constraint is the nonlocal analogue of a boundary condition necessary to demonstrate that the nonlocal diffusion equation is well-posed and is consistent with the jump process. …
A spectral approach to the narrow escape problem in the disk
本文利用准平稳分布和谱方法，对圆盘上具有小吸收窗口的布朗粒子窄逃逸问题，严格导出了第一特征值及特征函数法向导数的渐近估计，进而得到首次逃逸时间和逃逸点分布的渐近结果。
Exact Asymptotics for the Exit Time Probabilities of Scalar Ornstein-Uhlenbeck Bridges
本文研究了标量奥本-乌伦贝克桥的退出时间概率的精确渐进行为，通过幂级数方法提供了准确的估计，为评估OU过程在指定时间前逃离域的概率提供了理论支持。
In this paper we address the question of finding the point which maximizes the $p$-th moment of the exit time of planar Brownian motion from a given domain. We present a geometrical method of excluding parts of the domain from consideration which makes use of a coupling argument and the conformal invariance of Brownian motion. In many cases the maximizing point can be localized to a relatively small region. Several illustrative examples are presented.
The joint exit time and exit location law of a planar Ornstein-Uhlenbeck process on annular sectors and principal axis rectangles
本文研究平面Ornstein-Uhlenbeck过程在环形扇区和主轴矩形上的首次退出，通过本征函数展开获得退出时间与退出位置的联合定律，并用蒙特卡洛模拟验证，应用于光镊捕获胶体粒子。
We identify the critical exponent of integrability of the first exit time of rotation invariant stable Lévy process from parabola--shaped region.
First Passage and First Exit Times for diffusion processes related to a general growth curve
针对包含多种经典生长方程的一般生长曲线，提出加性、乘性噪声下的两种随机扩散模型，推导了时变阈值下首达时间与首出时间概率密度的闭式解。
The mean first exit (passage) time characterizes the average time of a stochastic process never leaving a fixed region in the state space, while the escape probability describes the likelihood of a transition from one region to another for a stochastic system driven by discontinuous (with jumps) Lévy motion. This paper discusses the two deterministic quantities, mean first exit time and escape probability, for the anomalous processes having the tempered Lévy stable waiting times with the tempering index $λ>0$ and the stability index $0<α\le 1$; as for the distribution of jump lengths or the ty…
Leftward, Rightward and Complete Exit Time Distributions of Jump Processes
该研究推导了对称跳跃过程在区间内向左、向右及完全退出时间分布的精确渐近表达式，揭示了由Levy指数控制的普适行为，填补了连续极限不适用时的理论空白。
American Options with Last Exit Times: A Free-Boundary Approach
本文研究了具有随机时间期限的美式看跌期权估值问题，通过扩展滤子和Azéma超martingale将问题转化为具有半连续时间依赖收益函数的最优停止问题，推导出非线性积分方程以刻画自由边界和价值函数。
Neuronal networks can generate burst events. It remains unclear how to analyse interburst periods and their statistics. We study here the phase-space of a mean-field model, based on synaptic short-term changes, that exhibit burst and interburst dynamics and we identify that interburst corresponds to the escape from a basin of attraction. Using stochastic simulations, we report here that the distribution of the these durations do not match with the time to reach the boundary. We further analyse this phenomenon by studying a generic class of two-dimensional dynamical systems perturbed by small n…
We review some of the theory relevant to passage times of one-dimensional Lévy processes out of bounded regions, highlighting results that are useful in physical phenomena modelled by heavy-tailed Lévy flights. The process is hypothesised to describe the motion of a particle on the line, starting at $0$, and exiting either a fixed interval $[-r, r]$, $r > 0$, or a time-dependent, expanding, set of intervals of the form $[-r t^κ, r t^κ]$, $r > 0$, $κ> 0$. Asymptotic behaviour of the exit time may be as $r \downarrow 0$ or as $r \to \infty$, but particular emphasis is placed herein on "small tim…
We present a study of the distance between a Brownian motion and a submanifold of a complete Riemannian manifold. We include a variety of results, including an inequality for the Laplacian of the distance function derived from a Jacobian comparison theorem, a characterization of local time on a hypersurface which includes a formula for the mean local time, an exit time estimate for tubular neighbourhoods and a concentration inequality. We derive the concentration inequality using moment estimates to obtain an exponential bound, which holds under fairly general assumptions and which is sufficie…
For a stopped diffusion process in a multidimensional time-dependent domain $\D$, we propose and analyse a new procedure consisting in simulating the process with an Euler scheme with step size $Δ$ and stopping it at discrete times $(iΔ)_{i\in\N^*}$ in a modified domain, whose boundary has been appropriately shifted. The shift is locally in the direction of the inward normal $n(t,x)$ at any point $(t,x)$ on the parabolic boundary of $\D$, and its amplitude is equal to $0.5826 (...) |n^*σ|(t,x)\sqrt Δ$ where $σ$ stands for the diffusion coefficient of the process. The procedure is thus extremel…
This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate the {\it stabilities} of the times, $\Tstarb(r)$ and $\Tbarb(r)$, at which $X$, started with $X_0=0$, first leaves the space-time regions $\{(t,y)\in\R^2: y\le rt^b, t\ge 0\}$ (one-sided exit), or $\{(t,y)\in\R^2: |y|\le rt^b, t\ge 0\}$ (two-sided exit), $0\le b<1$, as $r\dto 0$. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $…
A Discretization Scheme for BSDEs with Random Time Horizon
本文针对具有随机无界时间视界的Lipschitz BSDE，提出后向Euler离散格式并推导出由步长、终止时间及生成元时间分量逼近误差控制的强误差界，应用于有界域FBSDE时误差仅依赖退出时间距离。
Freidlin-Wentzell collision-laws between self-stabilizing diffusions
该研究针对双稳态势中独立自稳定扩散过程，适配Freidlin-Wentzell估计推导其零噪声极限下的近碰撞渐近行为，还拓展至平均场粒子系统近似及一维情形。
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