Markov process functionals in finance and insurance

The Markov property of Markov process functionals which are frequently used in economy, finance, engineering and statistic analysis is studied. The conditions to judge Markov property of some important Markov process functionals are presented, the following conclusions are obtained： the multidimensional process with independent increments is a multidimensional Markov process; the functional in the form of path integral of process with independent increments is a Markov process; the surplus process with the doubly stochastic Poisson process is a vector Markov process. The conditions for linear transformation of vector Markov process being still a Markov process are given.     

A new aspect of a risk process and its statistical inference

We introduce a new aspect of a risk process, which is a macro approximation of the flow of a risk reserve. We assume that the underlying process consists of a Brownian motion plus negative jumps, and that the process is observed at discrete time points. In our context, each jump size of the process does not necessarily correspond to the each claim size. Therefore our risk process is different from the traditional risk process. We cannot directly observe each jump size because of discrete observations. Our goal is to estimate the adjustment coefficient of our risk process from discrete observations.     

Ergodic and adaptive control of hidden Markov models

A partially observed stochastic system is described by a discrete time pair of Markov processes. The observed state process has a transition probability that is controlled and depends on a hidden Markov process that also can be controlled. The hidden Markov process is completely observed in a closed set, which in particular can be the empty set and only observed through the other process in the complement of this closed set. An ergodic control problem is solved by a vanishing discount approach. In the case when the transition operators for the observed state process and the hidden Markov process depend on a parameter and the closed set, where the hidden Markov process is completely observed, is nonempty and recurrent an adaptive control is constructed based on this family of estimates that is almost optimal.     

Pricing catastrophe options with counterparty credit risk in a reduced form model

In this paper, we study the price of catastrophe options with counterparty credit risk in a reduced form model. We assume that the loss process is generated by a doubly stochastic Poisson process, the share price process is modeled through a jump-diffusion process which is correlated to the loss process, the interest rate process and the default intensity process are modeled through the Vasicek model. We derive the closed form formulae for pricing catastrophe options in a reduced form model. Furthermore, we make some numerical analysis on the explicit formulae.     

Conditional law of risk processes given that ruin occurs

A risk process that can be Markovised is conditioned on ruin. We prove that the process remains a Markov process. If the risk process is a PDMP, it is shown that the conditioned process remains a PDMP. For many examples the asymptotics of the parameters in both the light-tailed case and the heavy-tailed case are discussed.     

违约风险的交换期权的定价模型与解法

含交易对手违约风险的交换期权采用混合模型定价,借助公司价值模型中的补偿率,同时采用以强度为基础的违约函数来确定违约的发生.假定违约强度遵从均值回复的重随机Poisson过程：且违约强度过程与标的资产,企业价值都相关.利用等价鞅测度变换方法导出含有违约风险的交换期权的价格闭解.     

Existence, uniqueness and ergodicity of Markov branching processes with immigration and instantaneous resurrection

We consider a modified Markov branching process incorporating with both state-independent immigration and instantaneous resurrection. The existence criterion of the process is firstly considered. We prove that if the sum of the resurrection rates is finite, then there does not exist any process. An existence criterion is then established when the sum of the resurrection rates is infinite. Some equivalent criteria, possessing the advantage of being easily checked, are obtained for the latter case. The uniqueness criterion for such process is also investigated. We prove that although there exist infinitely many of them, there always exists a unique honest process for a given q-matrix. This unique honest process is then constructed. The ergodicity property of this honest process is analysed in detail. We prove that this honest process is always ergodic and the explicit expression for the equilibrium distribution is established.     

Stochastic Processes with Age-Dependent Transition Rates

We study stochastic processes with age-dependent transition rates. A typical example of such a process is a semi-Markov process which is completely determined by the holding time distributions in each state and the transition probabilities of the embedded Markov chain. The process we construct generalizes semi-Markov processes. One important feature of this process is that unlike semi-Markov processes the transition probabilities of this process are age-dependent. Under certain condition we establish the Feller property of the process. Finally, we compute the limiting distribution of the process.     

Existence,uniqueness and ergodicity of Markov branching processes with immigration and instantaneous resurrection

We consider a modified Markov branching process incorporating with both state-independent immigration and instantaneous resurrection.The existence criterion of the process is firstly considered.We prove that if the sum of the resurrection rates is finite,then there does not exist any process.An existence criterion is then established when the sum of the resurrection rates is infinite.Some equivalent criteria,possessing the advantage of being easily checked,are obtained for the latter case.The uniqueness criterion for such process is also investigated.We prove that although there exist infinitely many of them,there always exists a unique honest process for a given q-matrix.This unique honest process is then constructed.The ergodicity property of this honest process is analysed in detail.We prove that this honest process is always ergodic and the explicit expression for the equilibrium distribution is established.     

On the problem of testing whether a mixed poisson process is homogeneous

In 2 recent paper Albrecht proposes a test procedure for testing whether a point process is homogeneous, also taking care of alternatives when the process is a mixed Poisson process. In the present note it is argued that it seems impossible to find a meaningful test criterion for testing whether a mixed Poisson process is homogeneous.     

A modified insurance risk process with uncertainty

An insurance risk process is traditionally considered by describing the claim process via a renewal reward process and assuming the total premium to be proportional to the time with a constant ratio. It is usually modeled as a stochastic process such as the compound Poisson process, and historical data are collected and employed to estimate the corresponding parameters of probability distributions. However, there exists the case of lack of data such as for a new insurance product. An alternative way is to estimate the parameters based on experts’ subjective belief and information. Therefore, it is necessary to employ the uncertain process to model the insurance risk process. In this paper, we propose a modified insurance risk process in which both the claim process and the premium process are assumed to be renewal reward processes with uncertain factors. Then we give the inverse uncertainty distribution of the modified process at each time. On this basis, we derive the ruin index which has an explicit expression based on given uncertainty distributions.     

Estimation for jump processes in the tangent bundle of a Riemann manifold

A stochastic process is formulated in the tangent bundle of a Riemann manifold where the vector fibre portion of the process is a jump process. Since the tangent spaces change as the process in the base manifold evolves, it is necessary to define a jump process in the fibres of the tangent bundle with respect to the process in the base manifold. An estimation problem is formulated and solved for a process obtained from the jump process in the fibres of the tangent bundle where the observations include the process in the base manifold and the jump times. Since each fibre of the tangent bundle is a linear space, a suitable modification of some results for estimation in linear spaces can be used to solve the aforementioned estimation problem.Research supported by NSF Grants ENG 75-06562 and MCS 76-01695 and AFOSR Grant 77-3177.     

An overview of Brownian and non-Brownian FCLTs for the single-server queue

We review functional central limit theorems (FCLTs) for the queue-content process in a single-server queue with finite waiting room and the first-come first-served service discipline. We emphasize alternatives to the familiar heavy-traffic FCLTs with reflected Brownian motion (RBM) limit process that arise with heavy-tailed probability distributions and strong dependence. Just as for the familiar convergence to RBM, the alternative FCLTs are obtained by applying the continuous mapping theorem with the reflection map to previously established FCLTs for partial sums. We consider a discrete-time model and first assume that the cumulative net-input process has stationary and independent increments, with jumps up allowed to have infinite variance or even infinite mean. For essentially a single model, the queue must be in heavy traffic and the limit is a reflected stable process, whose steady-state distribution can be calculated by numerically inverting its Laplace transform. For a sequence of models, the queue need not be in heavy traffic, and the limit can be a general reflected Lévy process. When the Lévy process representing the net input has no negative jumps, the steady-state distribution of the reflected Lévy process again can be calculated by numerically inverting its Laplace transform. We also establish FCLTs for the queue-content process when the input process is a superposition of many independent component arrival processes, each of which may exhibit complex dependence. Then the limiting input process is a Gaussian process. When the limiting net-input process is also a Gaussian process and there is unlimited waiting room, the steady-state distribution of the limiting reflected Gaussian process can be conveniently approximated. This revised version was published online in June 2006 with corrections to the Cover Date.     

The impact of data collection on fill rate performance in the (R,s, Q) inventory model

In this paper we consider the determination of the reorder point s in an (R, s, Q) inventory model subject to a fill rate service level constraint. We assume that the underlying demand process is a compound renewal process. We then derive an approximation method to compute the reorder level such that a target service level is achieved. Restrictions on the input parameters are given, within which this method is applicable. Moreover, we will investigate the effects on the fill rate performance in case the underlying demand process is indeed a compound renewal process, while the demand process is modelled as a discrete-time demand process. That is, the time axis is divided in time units (for example, days) and demands per time unit are independent and identically distributed random variables. It will be shown that smooth and erratic behaviour of the inter-arrival times have different impacts on the performance of the fill rate when demand is modelled as a discrete-time process and in case the underlying demand process is a compound renewal process.     

Bivariate step processes and random evolutions

A random evolution process constructed from regular step processes with a common state space and indexed on an evolution rule space is shown to be a regular step process on the product space. Conversely, it is shown that under mild conditions, any regular step process on a product space is equivalent to a random evolution process. Conditions are given on the cardinality of the spaces and on the parameters of the process that are sufficient for the process to have various recurrence and ergodicity properties. Applications to birth-death processes are given.     

A Cox Process Involved in the Bose–Einstein Condensation

The point process corresponding to the configurations of bosons in standard conditions is a Cox process driven by the square norm of a centered Gaussian process. This point process is infinitely divisible. We point out the fact that this property is preserved by the Bose–Einstein condensation phenomenon and show that the obtained point process after such a condensation occured, is still a Cox process but driven by the square norm of a shifted Gaussian process, the shift depending on the density of the particles. This law provides an illustration of a “super”- Isomorphism Theorem existing above the usual Isomorphism Theorem of Dynkin available for Gaussian processes. Submitted: February 8, 2008. Accepted: March 5, 2008.     

Ergodicity Conditions and Stationary Distributions of a Continuous Semi-Markov Process

A semi-Markov process of general type on a metric space is considered. The stationary distribution of the process is investigated. A Markov process associated with the semi-Markov one is constructed. The stationary distribution for values of the process at a fixed point of the time scale and for the lengths of intervals of constancy covering this point is expressed in terms of the stationary distribution of the associated Markov process. Bibliography: 11 titles.     

Departure Process of the MAP/SM/1 Queue

We study the departure process of a single server queue with Markovian arrival input and Markov renewal service time. We derive the joint transform of departure time and the number of departures and, based on this transform, we establish several expressions for burstiness (variance) and correlation (covariance sequence) of the departure process. These expressions reveal that burstiness and correlation of the arrival process have very little impact on the departure process when a queueing system is heavily loaded. In contrast, both burstiness and correlation of the service-time process greatly affect those of the departure process regardless of the load of the system. Finally, we show that, even when an arrival process is short-range dependent, the departure process could has long-range dependence if a service-time process is long-range dependent.     

关于随机过程循序可测性的一个注记

本文引进了可测空间的自然乘积空间和随机过程的自然提升过程等概念 .在此基础上 ,证明了一个随机过程循序可测的充分必要条件是其自然提升过程是适应过程 .此外 ,我们还得到其他一些有趣的结果 .     

Some Systems Aspects of Strategic Planning

This paper is concerned with the strategic planning process and the contribution that analysis can make to it. It develops a conceptual model of the strategic planning process based on the traditional control system model and identifies a number of basic elements of the process. Four groups of analytical tools are presented and their relevance to the planning process discussed. It is suggested that tools from the different groups may be used concurrently at different parts of the process, although the use of analytical tools in some parts of the planning process is not well established. Finally the role of the analyst in the overall planning process is discussed.