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.     

The construction of markov processes in random environments and the equivalence theorems

In sec.1, we introduce several basic concepts such as random transition function, p-m process and Markov process in random environment and give some examples to construct a random transition function from a non-homogeneous density function. In sec. 2, we construct the Markov process in random enviromment and skew product Markov process by p -m process and investigate the properties of Markov process in random environment and the original process and environment process and skew product process. In sec. 3, we give several equivalence theorems on Markov process in random environment.     

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.     

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.     

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.     

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.     

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.     

Dam processes with state dependent batch sizes and intermittent production processes with state dependent rates

We consider a dam process with a general (state dependent) release rule and a pure jump input process, where the jump sizes are state dependent. We give sufficient conditions under which the process has a stationary version in the case where the jump times and sizes are governed by a marked point process which is point (Palm) stationary and ergodic. We give special attention to the Markov and Markov regenerative cases for which the main stability condition is weakened. We then study an intermittent production process with state dependent rates. We provide sufficient conditions for stability for this process and show that if these conditions are satisfied, then an interesting new relationship exists between the stationary distribution of this process and a dam process of the type we explore here.Supported in part by The Israel Science Foundation, grant no. 372/93-1.     

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.     

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.     

Quantile Processes in the Presence of Auxiliary Information

We employ the empirical likelihood method to propose a modified quantile process under a nonparametric model in which we have some auxiliary information about the population distribution. Furthermore, we propose a modified bootstrap method for estimating the sampling distribution of the modified quantile process. To explore the asymptotic behavior of the modified quantile process and to justify the bootstrapping of this process, we establish the weak convergence of the modified quantile process to a Gaussian process and the almost-sure weak convergence of the modified bootstrapped quantile process to the same Gaussian process. These results are demonstrated to be applicable, in the presence of auxiliary information, to the construction of asymptotic bootstrap confidence bands for the quantile function. Moreover, we consider estimating the population semi-interquartile range on the basis of the modified quantile process. Results from a simulation study assessing the finite-sample performance of the proposed semi-interquartile range estimator are included.     

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.     

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.     

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.     

The empirical distribution function and partial sum process of residuals from a stationary arch with drift process

The weak convergence of the empirical process and partial sum process of the residuals from a stationary ARCH-M model is studied. It is obtained for and consistent estimate of the ARCH-M parameters. We find that the limiting Gaussian processes are no longer distribution free and hence residuals cannot be treated as i.i.d. In fact the limiting Gaussian process for the empirical process is a standard Brownian bridge plus an additional term, while the one for partial sum process is a standard Brownian motion plus an additional term. In the special case of a standard ARCH process, that is an ARCH process with no drift, the additional term disappears. We also study a sub-sampling technique which yields the limiting Gaussian processes for the empirical process and partial sum process as a standard Brownian bridge and a standard Brownian motion respectively.     

Spectral tail processes and max-stable approximations of multivariate regularly varying time series

A regularly varying time series as introduced in Basrak and Segers (2009) is a (multivariate) time series such that all finite dimensional distributions are multivariate regularly varying. The extremal behavior of such a process can then be described by the index of regular variation and the so-called spectral tail process, which is the limiting distribution of the rescaled process, given an extreme event at time 0. As shown in Basrak and Segers (2009), the stationarity of the underlying time series implies a certain structure of the spectral tail process, informally known as the “time change formula”. In this article, we show that on the other hand, every process which satisfies this property is in fact the spectral tail process of an underlying stationary max-stable process. The spectral tail process and the corresponding max-stable process then provide two complementary views on the extremal behavior of a multivariate regularly varying stationary time series.     

鞅方法在一类破产问题中的应用

本文从鞅条件出发 ,推导出了总理赔过程分别为复合 Poisson过程与复合二项过程 ,利率强度波动为带跳的 Poisson过程情形下的调节方程 ,并由此得到了一些有趣的结果。     

Walsh spectral analysis of multiple dyadic stationary processes and its applications

In this paper, we investigate some properties of multiple dyadic stationary processes from the viewpoint of their Walsh spectral analysis. It is shown that under certain conditions a dyadic autoregressive and moving average process of finite order is expressed as a dyadic autoregressive process of finite order and also as a dyadic moving average process of finite order. We can see that the principal component process of such a dyadic stationary process has a simple finite structure in the sense that a dyadic filter which generates the principal component process has only one-side finite lags.