Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries

We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class includes many interesting processes in real applications, e.g., Ornstein–Uhlenbeck, growth processes and geometric Brownian motion with time dependent drift. This method applies to both one-sided and two-sided general nonlinear boundaries, which may be discontinuous. Using this approach explicit formulas for boundary crossing probabilities for certain nonlinear boundaries are obtained, which are useful in evaluation and comparison of various computational algorithms. Moreover, numerical computation can be easily done by Monte Carlo integration and the approximation errors for general boundaries are automatically calculated. Some numerical examples are presented.      

A Monte Carlo Method for the Simulation of First Passage Times of Diffusion Processes

A reliable Monte Carlo method for the evaluation of first passage times of diffusion processes through boundaries is proposed. A nested algorithm that simulates the first passage time of a suitable tied-down process is introduced to account for undetected crossings that may occur inside each discretization interval of the stochastic differential equation associated to the diffusion. A detailed analysis of the performances of the algorithm is then carried on both via analytical proofs and by means of some numerical examples. The advantages of the new method with respect to a previously proposed numerical-simulative method for the evaluation of first passage times are discussed. Analytical results on the distribution of tied-down diffusion processes are proved in order to provide a theoretical justification of the Monte Carlo method.     

Boundary Crossing Probabilities of Jump Diffusion Processes to Time-Dependent Boundaries

Methodology and Computing in Applied Probability - The finite Markov chain imbedding technique has been used to compute the boundary crossing probabilities of one and higher-dimensional Brownian...     

Almost Sure Comparisons for First Passage Times of Diffusion Processes through Boundaries

Conditions on the boundary and parameters that produce ordering in the first passage time distributions of two different diffusion processes are proved making use of comparison theorems for stochastic differential equations. Three applications of interest in stochastic modeling are presented: a sensitivity analysis for diffusion models characterized by means of first passage times, the comparison of different diffusion models where first passage times represent an important feature and the determination of upper and lower bounds for first passage time distributions.     

离散时间马氏决策过程的首达目标准则

本文考虑可数状态离散时间马氏决策过程的首达目标模型的风险概率准则.优化的准则是最小化系统首次到达目标状态集的时间不超过某阈值的风险概率.首先建立最优方程并且证明最优值函数和最优方程的解对应,然后讨论了最优策略的一些性质,并进一步给出了最优平稳策略存在的条件,最后用一个例子说明我们的结果.     

Stochastic Dynamics for Passage Times and Diffusion Approximations for Finite Capacity Storage Models with Different Kinds of Barriers

In this article, we discuss a number of storage models of finite capacity with random inputs, random outputs, and linear release policy. They form a class of one-dimensional master equations with separable kernels. For this class of problems, the integral equations for first overflow or first emptiness can be transformed exactly into ordinary differential equations. Analysis is done with separable kernel. For all the stochastic models, two barriers are considered: one at X = 0 and the other at X = k, and the barriers are treated as absorbing or reflecting. The imbedding method is used to derive a third order differential equation. We consider first passage times for overflow without or with emptiness of the dam. We also study the passage times for first emptiness with and without overflows. The expected amount of overflows in a given time is also calculated. Finally, by suitable statistical features, all these models are converted into diffusion process with drift. Closed form solutions are obtained for all the problems in terms of Laplace transform functions. For the diffusion process with drift first passage time density is arrived at by treating X = 0 and X = k as absorbing barriers. One of the barriers as reflecting is also studied.     

Boundary Crossing for the Difference of Two Ordinary or Compound Poisson Processes

We consider the lower boundary crossing problem for the difference of two independent compound Poisson processes. This problem arises in the busy period analysis of single-server queueing models with work removals. The Laplace transform of the crossing time is derived as the unique solution of an integral equation and is shown to be given by a Neumann series. In the case of ±1 jumps, corresponding to queues with deterministic service times and work removals, we obtain explicit results and an approximation useful for numerical purposes. We also treat upper boundaries and two-sided stopping times, which allows to derive the conditional distribution of the maximum workload up to time t, given the busy period is longer than t.     

R~n上扩散过程的代数式收敛

本文研究n维欧氏空间上的扩散过程在L2意义下的代数式收敛的情况,给出了判定代数式收敛的方法,并对两种特殊情形扩散算子进行了讨论.将所得判敛法应用于两个例子可得到精确的结果.     

Sample Path Properties of a Class of Operator Stable Processes

Abstract

Let X = {X(t), t ? ?₊} be an operator stable Lévy process on ? ^d with the exponent B, where B is a diagonal matrix. In the present paper, we consider the asymptotic behavior of the first passage time out of a sphere, and of the sojourn time in a sphere. We shall also determine the exact Hausdorff measure function for the range of X over unit time interval [0, 1].     

扩散过程代数式收敛定性的判别准则

本文定性地讨论非紧空间中可逆扩散过程的代数式收敛的判定 .使用分裂空间的方法 .将全空间分裂成两个部分 :紧的子空间与非紧的余子空间 .在紧子空间中考虑边界反射的Neumann过程 ,它必然是代数式收敛的 .而在非紧子空间中考虑边界吸收的Dirichlet过程 ,如果这一Dirichlet过程以代数式的速度击中边界 ,那么就有原过程在全空间代数式收敛 ;反之 ,原过程代数式收敛 ,非紧子空间中的Dirichlet过程也是代数式收敛的 .因此过程在紧子空间的任意摄动不会影响在全空间的代数式收敛性 .     

标准布朗运动关于曲线边界通过概率

布朗运动是一种重要的随机过程,它的首出时的分布在很多方面有着重要的应用.该文讨论了布朗运动关于任意曲线边界的首出时的问题,求出了布朗运动停在双侧(单侧)曲线边界内的概率的分析表达式.     

标准布朗运动关于线性边界通过概率

该文讨论了布朗运动关于线性边界的首出时问题,求出了布朗运动停留在双侧(单侧)逐段线性边界内的概率的分析表达式.     

Transition Probabilities for Symmetric Jump Processes

We consider symmetric Markov chains on the integer lattice in dimensions, where and the conductance between and is comparable to . We establish upper and lower bounds for the transition probabilities that are sharp up to constants.

     

具有不良影响的一类存贮过程的首中时

考虑一类具有正负跳(正负跳大小服从Erlang分布)的存贮过程的首中时,利用马氏无穷小算子的方法来刻画首中时的拉普拉斯变换.     

一维扩散过程的小随机扰动

本文考虑一维扩散过程的小随机扰动．我们应用随机分析方法，给出了当扰动趋于零时，平均越出时间的渐近估计和越出时间的概率估计．     

Sharp Error Estimate for Maximum Likelihood Estimator of Nonstationary Diffusion Processes

We consider the maximum likelihood estimator of the unknown parameter in a class of nonstationary diffusion processes. We give further a precise estimate for the error of the estimator.     

First passage probabilities of one-dimensional diffusion processes

This work is devoted to calculating the first passage probabilities of one-dimensional diffusion processes. For a one-dimensional diffusion process, we construct a sequence of Markov chains so that their absorption probabilities approximate the first passage probability of the given diffusion process. This method is especially useful when dealing with time-dependent boundaries.     

On the Asymptotic Behavior of First Passage Time Densities for Stationary Gaussian Processes and Varying Boundaries

Making use of a Rice-like series expansion, for a class of stationary Gaussian processes the asymptotic behavior of the first passage time probability density function through certain time-varying boundaries, including periodic boundaries, is determined. Sufficient conditions are then given such that the density asymptotically exhibits an exponential behavior when the boundary is either asymptotically constant or asymptotically periodic.     

Poisson Broken Lines Process and its Application to Bernoulli First Passage Percolation

In this note we introduce a process, which we call 'the Poisson broken lines process", and we compute the intensity of a point process which is obtained by intersecting the Poisson broken lines process with an abscissa axis. In the second part we apply this result to compute an explicit lower bound for the time constant of a planar Bernoulli first passage percolation model with the parameter p < p_c.     

Doubling-Time Probability Densities For Growth Processes

The doubling-time probability density of a growth process is the probability density for the time it takes for the size to double. Doubling-time probability densities are useful in studying growth rates, for example, of organisms, populations, financial products, or chemical reactions. Three fundamental stochastic models of growth are investigated for their doubling-time probability densities. It is shown that two of the stochastic models have doubling-time probability densities which are inverse Gaussian. Although the third stochastic model’s doubling-time density does not have a simple analytical form, it is shown to be approximately inverse Gaussian under a reasonable hypothesis on the model’s parameters. Two data sets for doubling time, spruce seedling size and Texas Mega Millions Lottery jackpot, are fit to inverse Gaussian distributions.