Potentials and the Distributions of the Last Exit Times of Birth and Death Processes

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The asymptotic relation between the first crossing point and the last exit time of Gaussian order statistics sequences

In this paper,we study the asymptotic relation between the first crossing point and the last exit time for Gaussian order statistics which are generated by stationary weakly and strongly dependent Gaussian sequences.It is shown that the first crossing point and the last exit time are asymptotically independent and dependent for weakly and strongly dependent cases,respectively.The asymptotic relations between the first crossing point and the last exit time for stationary weakly and strongly dependent Gaussian sequences are also obtained.

AN ESTIMATE ON THE DISTRIBUTION AND MOMENTS OF THE LAST EXIT TIME OF AN ELLIPTIC DIFFUSION PROCESS

Let Ls be the last exit time from a compact set B of an elliptic diffusion process X. A moderate estimate for the distribution of Ls is obtained, and the sufficient and necessary condition for E^x（L^κB） 〈∞ is proved.

Extraction Phase Simulation of Cargo Airdrop System

To study the characteristics of cargo extraction, the initial phase of airdrop process, a high fidelity and extendibility simulation model with uniform motion equations for all states during extraction is developed on the basis of dynamics methods and contact models between cargo and aircraft. Simulation results agree well with tests data. Cargo exit parameters, which contribute to cargo pitch after extraction, are studied. Simplified computation model of dimensionless exit time is developed and used to evaluate the relation between extraction phase and landing accuracy. Safe interval model is introduced to evaluate the safety of extraction process. Also, relations between initial parameters, including pull coefficient, aircraft pitch and CG coefficient, etc, and result parameters, including exit time, cargo safety, pitch, etc, are developed to help design of airdrop system, especially the selection of extraction parachute and cargo deployment.

A Joint Laplace Transform for Pre-exit Diffusion of Occupation Times

For a < r < b, the approach of Li and Zhou(2014) is adopted to find joint Laplace transforms of occupation times over intervals(a, r) and(r, b) for a time homogeneous diffusion process before it first exits from either a or b. The results are expressed in terms of solutions to the differential equations associated with the diffusions generator. Applying these results, we obtain more explicit expressions on the joint Laplace transforms of occupation times for Brownian motion with drift, Brownian motion with alternating drift and skew Brownian motion, respectively.

BEHAVIOR OF THE GLAUBER DYNAMICS IN THE LIMIT AS THE TEMPERATURE GOES TO ZERO

In this paper, we generalize Freidlin and Wentzell machinery [7] to the case of Glauber dynamics, which is still a stochastic dynamics even as the temperature vanishes. We first consider the probability to exit from an attractive basin and enter into another one in the limit as the temperature goes to zero. The first exiting time to escape from an attractive basin of an attractor in which the process starts is estimated. Then we show conclusions above in the case of exiting from a region formed by geveral attractive basins. The unpredictability of the exiting time is proved.

The Joint Distribution of the Maximum Excursion and the Minimum Excursion for Brownian Motion with Drift

In this paper, we discuss the problem of extreme value for Brownian motion with positive drift. We obtain the joint distribution of the maximum excursion and the minimum excursion.

Nonlocal Dynamics for Non-Gaussian Systems Arising in Biophysical Modeling

The aim of this article is to review our recent work on nonlocal dynamics of non-Gaussian systems arising in a gene regulatory network.We have used the mean exit time,escape probability and maximal likely trajectory to quantify dynamical behaviors of a stochastic diferential system with non-Gaussianα-stable Lévy motions,to examine how the nonGaussianity index and noise intensity afect the gene transcription processes.

On Exit Time from Balls of Jump-type Symmetric Markov Processes

We obtain upper and lower bounds of the exit times from balls of a jump-type symmetric Markov process. The proofs are delivered separately. The upper bounds are obtained by using the Levy system corresponding to the process, while the precise expression of the （L^2-）generator of the Dirichlet form associated with the process is used to obtain the lower bounds.

Portfolio Optimization with Uncertain Exit Time in Infinite-Time Horizon

In this paper, we study infinite-period mean-variance formulations for portfolio selections with an uncertain exit time. We employ the convergence control method together with the dynamic programming algorithm to derive analytical expressions for the optimal portfolio policy and the mean-variance efficient frontier under certain conditions. We illustrate these results by an numerical example.

On the Mean Exit Time for Compact Symmetric Spaces

Given a compact symmetric space, M, we obtain the mean exit time function from a principal orbit, for a Brownian particle starting and moving in a generalized ball whose boundary is the principal orbit. We also obtain the mean exit time flmction of a tube of radius r around special totally geodesic submanifolds P of M. Finally we give a comparison result for the mean exit time function of tubes around submanifolds in Riemannian manifolds, using these totally geodesic submanifolds in compact symmetric spaces as a model.

On First Returning Time and Last Exit Time of a Class of Markov Chain

Let {Xn} be a Markov chain with transition probability pij =： aj-（i-1）＋,i,j ≥ 0, where aj=0 providedj 〈 0, a0 〉 0, a0＋a1〈 1 and ∑∞n=0 an= 1. Let μ∑∞n=1nan. It is known that {Xn} is positive recurrent when μ 〈 1; is null recurrent when μ= 1; and is transient when μ 〉 1. In this paper, the integrability of the first returning time and the last exit time are discussed. Keywords Geom/G/1 queuing model, first returning time, last exit time, Markov chain

Exit problems for nonlinear stochastic evolution equations on Hilbert spaces

This paper extends exit theorems of Da Prato and Zabczyk to nonconstant diffusion coefficients.It uses extensively general, exponential estimates due to Peszat.

Variational Principles for Asymptotic Variance of General Markov Processes

A variational formula for the asymptotic variance of general Markov processes is obtained.As application,we get an upper bound of the mean exit time of reversible Markov processes,and some comparison theorems between the reversible and non-reversible diffusion processes.

Small Random Perturbations of One-Dimensional Diffusion Processes

In the present paper we consider the small random perturbations of one-dimensional diffosion processes. By virtue of stochastic analysis methods, we investigate the asymptotics of the mean exit times and probabilistical estimates of the exit times as the perturbations tend to zero.

An occupation time related potential measure for diffusion processes

In this paper, for homogeneous diffusion processes, the approach of Y. Li and X. Zhou [Statist. Probab. Lett., 2014, 94： 48-55] is adopted to find expressions of potential measures that are discounted by their joint occupation times over semi-infinite intervals （-∞, a） and （a, ∞）. The results are expressed in terms of solutions to the differential equations associated with the diffusions generator. Applying these results, we obtain more explicit expressions for Brownian motion with drift, skew Brownian motion, and Brownian motion with two-valued drift, respectively.

The Weighted Transience and Recurrence of Markov Processes

Transience and recurrence are among the most important concepts in Markov processes. In this paper, we study the transience and recurrence for right processes with a given weight function, and characterize them by potentials, excessive functions, first hitting times and last exit times of the process. We also study the properties of recurrent states.

A FIRST-ORDER NUMERICAL SCHEME FOR FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS IN BOUNDED DOMAINS

We propose a novel numerical scheme for decoupled forward-backward stochastic differ- ential equations （FBSDEs） in bounded domains, which corresponds to a class of nonlinear parabolic partial differential equations with Dirichlet boundary conditions. The key idea is to exploit the regularity of the solution （Yt,Zt） with respect to Xt to avoid direct ap- proximation of the involved random exit time. Especially, in the one-dimensional case, we prove that the probability of Xt exiting the domain within At is on the order of O（（△t）ε exp（--1/（△t）2ε））, if the distance between the start point X0 and the boundary is 1 g at least on the order of O（△t）^1/2-ε ） for any fixed c 〉 0. Hence, in spatial discretization, we set the mesh size △x - （9（（At）^1/2-ε ）, so that all the interior grid points are sufficiently far from the boundary, which makes the error caused by the exit time decay sub-exponentially with respect to △t. The accuracy of the approximate solution near the boundary can be guaranteed by means of high-order piecewise polynomial interpolation. Our method is developed using the implicit Euler scheme and cubic polynomial interpolation, which leads to an overall first-order convergence rate with respect to △t.

Total Duration of Negative Surplus for a Brownian Motion Risk Model with Interest

In this paper,we consider the Brownian motion risk model with interest.The Laplace transform of the first exit time from the upper barrier before hitting the lower barrier is obtained.Using the obtained result and exploiting the limitation idea,we derive the Laplace transform of total duration of negative surplus.