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.     

First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes

We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original boundary by a different one. In doing so we establish the existence of the first-passage time density and provide an upper bound for this function. In the case of processes with diffusion interval equal to ℝ this is extended to a lower bound, as well as bounds for the first crossing time of a lower boundary. An extension to some time-inhomogeneous diffusions is given. These results are illustrated by numerical examples.      

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.     

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.      

允许赤字的MAP风险过程首达时

保险公司作为负债经营的特殊企业,其偿付能力受到监管部门的约束,本文以公司负债经营为前提研究其各种首次时.考虑MAP风险过程,即存在一随机背景Markov过程,索赔到达与索赔大小同时受这一背景过程影响,索赔到达为Markov到达点过程(MAP),索赔大小对于不同的背景状态具有不同的分布.本文给出首达时满足的积分-微分方程,通过求解带边界条件的积分-微分方程,给出了盈余过程从初始盈余水平到达某一给定盈余水平的首达时的Laplace变换的矩阵表示式,并由此推得了盈余过程到达指定水平的若干首达事件概率.     

A Diffusion Approximation for a Markovian Queue with Reneging

Consider a single-server queue with a Poisson arrival process and exponential processing times in which each customer independently reneges after an exponentially distributed amount of time. We establish that this system can be approximated by either a reflected Ornstein–Uhlenbeck process or a reflected affine diffusion when the arrival rate exceeds or is close to the processing rate and the reneging rate is close to 0. We further compare the quality of the steady-state distribution approximations suggested by each diffusion.     

Directionally Convex Comparison of Correlated First Passage Times

Many important classes of multivariate distributions arising from reliability modeling are the distributions of correlated first passage times of certain multivariate point processes. In this paper, we obtain results that compare variability and dependence structure of these correlated first passage times, in the sense of directionally convex ordering. Under certain conditions, we also obtain some easily computable distributional bounds for the first passage times whose joint distributions can not be expressed explicitly.     

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.     

Shrinkage drift parameter estimation for multi‐factor Ornstein–Uhlenbeck processes

We consider some inference problems concerning the drift parameters of multi‐factors Vasicek model (or multivariate Ornstein–Uhlebeck process). For example, in modeling for interest rates, the Vasicek model asserts that the term structure of interest rate is not just a single process, but rather a superposition of several analogous processes. This motivates us to develop an improved estimation theory for the drift parameters when homogeneity of several parameters may hold. However, the information regarding the equality of these parameters may be imprecise. In this context, we consider Stein‐rule (or shrinkage) estimators that allow us to improve on the performance of the classical maximum likelihood estimator (MLE). Under an asymptotic distributional quadratic risk criterion, their relative dominance is explored and assessed. We illustrate the suggested methods by analyzing interbank interest rates of three European countries. Further, a simulation study illustrates the behavior of the suggested method for observation periods of small and moderate lengths of time. Our analytical and simulation results demonstrate that shrinkage estimators (SEs) provide excellent estimation accuracy and outperform the MLE uniformly. An over‐ridding theme of this paper is that the SEs provide powerful extensions of their classical counterparts. Copyright © 2009 John Wiley & Sons, Ltd.     

Non-Differentiable Skew Convolution Semigroups and Related Ornstein–Uhlenbeck Processes

It is proved that a general non-differentiable skew convolution semigroup associated with a strongly continuous semigroup of linear operators on a real separable Hilbert space can be extended to a differentiable one on the entrance space of the linear semigroup. A càdlàg strong Markov process on an enlargement of the entrance space is constructed from which we obtain a realization of the corresponding Ornstein–Uhlenbeck process. Some explicit characterizations of the entrance spaces for special linear semigroups are given.     

Properties of the Reflected Ornstein–Uhlenbeck Process

Consider an Ornstein–Uhlenbeck process with reflection at the origin. Such a process arises as an approximating process both for queueing systems with reneging or state-dependent balking and for multi-server loss models. Consequently, it becomes important to understand its basic properties. In this paper, we show that both the steady-state and transient behavior of the reflected Ornstein–Uhlenbeck process is reasonably tractable. Specifically, we (1) provide an approximation for its transient moments, (2) compute a perturbation expansion for its transition density, (3) give an approximation for the distribution of level crossing times, and (4) establish the growth rate of the maximum process.     

First passage time for some stationary processes

For a 1-dependent stationary sequence {X_n} we first show that if u satisfies p₁=p₁(u)=P(X₁>u)0.025 and n>3 is such that 88np₁³1, then

P{max(X₁,…,X_n)u}=ν·μⁿ+O{p₁³(88n(1+124np₁³)+561)}, n>3,

where

ν=1−p₂+2p₃−3p₄+p₁²+6p₂²−6p₁p₂,μ=(1+p₁−p₂+p₃−p₄+2p₁²+3p₂²−5p₁p₂)⁻¹

with

p_k=p_k(u)=P{min(X₁,…,X_k)>u}, k1

and

|O(x)||x|.

From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Y_s}, a similar, in terms of P{max_0skTY_s<u}, k=1,2 formula for P{max_0stY_su}, t>3T and apply this formula to the process Y_s=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.     

On a Class of Non-Translation Invariant Feller Semigroups on Lie Groups

We consider a class of Feller semigroups on Lie groups which fail to commute with left translation due to the existence of a cocycle h which is identically one for Lévy processes. Under certain conditions, we are able to show that the infinitesimal generator of such a semigroup has the Lévy–Khintchine–Hunt form but with variable characteristics, thus we obtain an extension of classical work in Euclidean space by Courrège.     

Central and Functional Central Limit Theorems for a Class of Urn Models

We consider an approach based on tails to certain central limit and functional central limit theorems for a class of two color urn models. In particular, some of the results are derived from an associated Ornstein–Uhlenbeck process, and for another result we give an alternative proof based on martingale tails.      

Generalized Mehler Semigroups and Catalytic Branching Processes with Immigration

Skew convolution semigroups play an important role in the study of generalized Mehler semigroups and Ornstein–Uhlenbeck processes. We give a characterization for a general skew convolution semigroup on a real separable Hilbert space whose characteristic functional is not necessarily differentiable at the initial time. A connection between this subject and catalytic branching superprocesses is established through fluctuation limits, providing a rich class of non-differentiable skew convolution semigroups. Path regularity of the corresponding generalized Ornstein–Uhlenbeck processes in different topologies is also discussed.     

Generalized z-Distributions and Related Stochastic Processes

The class of generalized z–distributions is defined and their properties are investigated. Ornstein–Uhlenbeck–type and self–similar generalized z–processes are constructed and described. Esscher transforms of the generalized z–processes and the mixed generalized z–processes are characterized. Finally, construction and some properties of generalized z–diffusions are also discussed.     

An asymptotic test for quantitative gene detection

The problem of detecting the presence of a quantitative gene using a great number of markers in a backcross genetic scheme is addressed.An asymptotic test based on the maximum of a differentiable stochastic process is constructed. Bounds for threshold and power calculation are presented. Simulations and numerical experiments illustrate the convergence towards the asymptotic distribution and the sharpness of the bounds.     

Almost Sure Convergence of the Numerical Discretization of Stochastic Jump Diffusions

Based on the shuffle product expansion of exponential Lie series in terms of a Philip Hall basis for the stochastic differential equations of jump-diffusion type, we can establish Stratonovich–Taylor–Hall (STH) schemes. However, the STHr scheme converges only at order r in the mean-square sense. In order to have the almost sure Stratonovich–Taylor–Hall (ASTH) schemes, we have to include all the terms related to multiple Poissonian integrals as the moments of multiple Poissonian integrals always have lower orders of magnitudes as compared with those of multiple Brownian integrals.     

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.     

关于多维扩散过程耦合的保序性

本文给出了多维扩散过程耦合保序的若干充分条件和必要条件，特别地，当耦合过程存在唯一时，得到了耦合保序的充要条件。