On the Transition Density and First Hitting Time Distributions of the Doubly Skewed CIR Process

Methodology and Computing in Applied Probability - In this paper, we study doubly skewed CIR processes, which are extensions of skew Brownian motion. We use modified spectral expansion to obtain...     

Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion

A large deviations type approximation to the probability of ruin within a finite time for the compound Poisson risk process perturbed by diffusion is derived. This approximation is based on the saddlepoint method and generalizes the approximation for the non-perturbed risk process by Barndorff-Nielsen and Schmidli (Scand Actuar J 1995(2):169–186, 1995). An importance sampling approximation to this probability of ruin is also provided. Numerical illustrations assess the accuracy of the saddlepoint approximation using importance sampling as a benchmark. The relative deviations between saddlepoint approximation and importance sampling are very small, even for extremely small probabilities of ruin. The saddlepoint approximation is however substantially faster to compute.     

Some Results for the Compound Poisson Process That Is Perturbed by Diffusion

Abstract In the present paper surplus process perturbed by diffusion are considered.The distributions ofthe surplus immediately before and at ruin corresponding to the probabilities of ruin caused by oscillation andruin caused by a claim are studied.Some joint distribution densities are obtained.Techniques from martingaletheory and renewal theory are used.     

First Hitting Time Analysis of the Independence Metropolis Sampler

In this paper, we study a special case of the Metropolis algorithm, the Independence Metropolis Sampler (IMS), in the finite state space case. The IMS is often used in designing components of more complex Markov Chain Monte Carlo algorithms. We present new results related to the first hitting time of individual states for the IMS. These results are expressed mostly in terms of the eigenvalues of the transition kernel. We derive a simple form formula for the mean first hitting time and we show tight lower and upper bounds on the mean first hitting time with the upper bound being the product of two factors: a “local” factor corresponding to the target state and a “global” factor, common to all the states, which is expressed in terms of the total variation distance between the target and the proposal probabilities. We also briefly discuss properties of the distribution of the first hitting time for the IMS and analyze its variance. We conclude by showing how some non-independence Metropolis–Hastings algorithms can perform better than the IMS and deriving general lower and upper bounds for the mean first hitting times of a Metropolis–Hastings algorithm.     

Asymptotics for the Moments of the Time to Ruin for the Compound Poisson Model Perturbed by Diffusion

We obtain the asymptotic behaviour of the k-th moment of the time to ruin in the classical risk model perturbed by diffusion for the case where the claim size distribution has a heavy tail.     

复合泊松过程的可加性

对复合泊松分布可加性的研究在许多的文献中都可以看到,本文首先应用特征函数的方法证明了复合泊松分布的可加性.以此为基础,结合对随机过程相关性质的讨论,证明了复合泊松过程也具有与复合泊松分布可加性相似的,某种意义上的可加性性质.     

带干扰的复合广义齐次Poisson过程的多险种破产概率

将复合广义齐次poisson过程的多险种风险模型推广到带干扰的一种新模型,运用鞅方法破产概率满足的Lundberg不等式和一般公式.     

带干扰的双复合Poisson风险模型

对古典风险模型进行推广,主要研究保费收入过程为带干扰双复合Poisson过程的风险模型,运用鞅的方法得出了破产概率满足的Lundburg不等式.     

Value at Ruin and Tail Value at Ruin of the Compound Poisson Process with Diffusion and Efficient Computational Methods

We analyze the insurer risk under the compound Poisson risk process perturbed by a Wiener process with infinite time horizon. In the first part of this article, we consider the capital required to have fixed probability of ruin as a measure of risk and then a coherent extension of it, analogous to the tail value at risk. We show how both measures of risk can be efficiently computed by the saddlepoint approximation. We also show how to compute the stabilities of these measures of risk with respect to variations of probability of ruin. In the second part of this article, we are interested in the computation of the probability of ruin due to claim and the probability of ruin due to oscillation. We suggest a computational method based on upper and lower bounds of the probability of ruin and we compare it to the saddlepoint and to the Fast Fourier transform methods. This alternative method can be used to evaluate the proposed measures of risk and their stabilities with heavy-tailed individual losses, where the saddlepoint approximation cannot be used. The numerical accuracy of all proposed methods is very high and therefore these measures of risk can be reliably used in actuarial risk analysis.     

Exact Asymptotics for the Distribution of the Time of Attaining the Maximum for a Trajectory of a Compound Poisson Process with Linear Drift

We consider the random process at − v+(pt) + v−(−qt), t ∈ (−∞, −), where v− and v+ are independent standard Poisson processes if t ≥ 0 and v−(t) = v+(t) = 0 if t < 0. Under certain conditions on the parameters a, p, and q, we study the distribution function G = G(x) of the time of attaining the maximum for a trajectory of this process. In the present article, we find an exact asymptotics for the tails of G. We also find a connection between this problem and the statistical problem of estimation of an unknown discontinuity point of a density function.     

Some Explicit Results on First Exit Times for a Jump Diffusion Process Involving Semimartingale Local Time

Journal of Theoretical Probability - In this paper, we consider the one-sided and the two-sided first exit problem for a jump diffusion process with semimartingale local time. Denote this process...     

On First Hitting Times for Skew CIR Processes

In this work, the first hitting times for skew CIR processes are investigated. We compute the Laplace transforms and the means of the first hitting times of some given levels. The solutions for Laplace transforms are in terms of Tricomi and Kummer confluent hypergeometric functions. We also exhibit the hitting time densities numerically at the end of this paper.     

On the First Exit Time of a Completely Asymmetric Stable Process from a Finite Interval

We compute the Laplace transform of the distribution of thefirst exit time from a finite interval for a completely asymmetricstable process. The formula involves a Mittag-Leffler functionand its derivative. As an application, we determine the asymptotictail behaviour of the foregoing distribution, and deduce anextension of the law of the iterated logarithm of Chung.     

Number of Jumps in Two-Sided First-Exit Problems for a Compound Poisson Process

In this paper, we study the joint Laplace transform and probability generating functions of two pairs of random variables: (1) the two-sided first-exit time and the number of claims by this time; (2) the two-sided smooth exit-recovery time and its associated number of claims. The joint transforms are expressed in terms of the so-called doubly-killed scale function that is defined in this paper. We also find explicit expressions for the joint density function of the two-sided first-exit time and the number of claims by this time. Numerical examples are presented for exponential claims.     

First Hitting Time of Brownian Motion on Simple Graph with Skew Semiaxes

Consider a stochastic process that lives on n-semiaxes emanating from a common origin. On each semiaxis it behaves as a Brownian motion and at the origin it chooses a semiaxis randomly. In this paper we study the first hitting time of the process. We derive the Laplace transform of the first hitting time, and provide the explicit expressions for its density and distribution functions. Numerical examples are presented to illustrate the application of our results.

     

复合泊松过程及其在保险风险中的若干应用

本文利用齐次泊松过程的可加性,研究了复合泊松过程的可加性及其性质。作为应用,讨论了单个理赔额服从指数分布的复合泊松风险模型在第n次索赔时发生负盈余的概率。     

First Exit Times for Ordinary and Compound Poisson Processes with Non-linear Boundaries

Distributions of the first-exit times from a region with concave upper boundary are discussed for ordinary and compound Poisson processes. Explicit formulae are developed for the case of ordinary Poisson processes. Recursive formulae are given for the compound Poisson case, where the jumps are positive, having discrete or continuous distributions with finite means. Applications to sequential point estimation and insurance are illustrated.      

Systems of stochastic Poisson equations: Hitting probabilities

We consider a $d$-dimensional random field $u=(u\left(x\right),x\in D)$ that solves a system of elliptic stochastic equations on a bounded domain $D?{\mathbb{R}}^k$, with additive white noise and spatial dimension $k=1,2,3$. Properties of $u$ and its probability law are proved. For Gaussian solutions, using results from Dalang and Sanz-Solé (2009), we establish upper and lower bounds on hitting probabilities in terms of the Hausdorff measure and Bessel–Riesz capacity, respectively. This relies on precise estimates of the canonical distance of the process or, equivalently, on $L^2$ estimates of increments of the Green function of the Laplace equation.