一个风险模型的研究

研究了索赔到达过程为平稳无后效流,保单到达过程为平稳无后效流,并带扩散扰动项的盈余过程．讨论了该盈余过程的马尔科夫性和鞅性．然后用鞅方法得到其破产概率的表达式及其相应的Lundberg不等式．     

Predictors for stationary processes with finite generalized Markovian property

A theorem is proved which establishes the conditions for a Gaussian vector stationary process to be Markovian. For a stationary process with finite generalized Markov property we construct a vector Markov process whose first coordinate coincides with the given process. Applying our theorem to the vector process, we derive formulas for the linear predictor of a process with finite generalized Markov property.Translated from Statisticheskie Metody, pp. 82–90, 1980.I would like to acknowledge the helpful attention of P. N. Sapozhnikov.     

Birthing Markov processes at random rates

Summary This paper studies processes constructed by birthing the trajectories of a given Markov process along time according to random probabilities. Getoor has considered the case where the random probabilities are determined by comultiplicative functionals and proved for right processes that the post-birth process has the Markov property. Here randomizations of comultiplicative functionals are described which give rise to conditionally Markov processes. The main argument is developed for general Markov processes and the transition probabilities of the new process, including those from the pre-birth state, are explicited.     

On risk processes with the Markov property and with independent increments

We study the influence on the underlying counting process of the Markov property and of the property of independent increments for a risk process.     

Ruin probabilities for a risk model with two classes of claims

In this paper we consider a risk model with two kinds of claims, whose claims number processes are Poisson process and ordinary renewal process respectively. For this model, the surplus process is not Markovian, however, it can be Markovianized by introducing a supplementary process, We prove the Markov property of the related vector processes. Because such obtained processes belong to the class of the so-called piecewise-deterministic Markov process, the extended infinitesimal generator is derived, exponential martingale for the risk process is studied. The exponential bound of ruin probability in iafinite time horizon is obtained.     

Integral-type functionals of first hitting times for continuous-time Markov chains

We investigate integral-type functionals of the first hitting times for continuous-time Markov chains. Recursive formulas and drift conditions for calculating or bounding integral-type functionals are obtained. The connection between the subexponential integral-type functionals and the subexponential ergodicity is established. Moreover, these results are applied to the birth-death processes. Polynomial integral-type functionals and polynomial ergodicity are studied, and a sufficient criterion for a central limit theorem is also presented.     

The age of a Markov process

The concept of a limiting conditional age distribution of a continuous time Markov process whose state space is the set of non-negative integers and for which {0} is absorbing is defined as the weak limit as t→∞ of the last time before t an associated “return” Markov process exited from {0} conditional on the state, j, of this process at t. It is shown that this limit exists and is non-defective if the return process is ρ-recurrent and satisfies the strong ratio limit property. As a preliminary to the proof of the main results some general results are established on the representation of the ρ-invariant measure and function of a Markov process. The conditions of the main results are shown to be satisfied by the return process constructed from a Markov branching process and by birth and death processes. Finally, a number of limit theorems for the limiting age as j→∞ are given.     

HILBERTIAN INVARIANCE PRINCIPLE FOR EMPIRICAL PROCESS ASSOCIATED WITH A MARKOV PROCESS

The authors establish the Hilbertian invariance principle for the empirical process of a stationary Markov process, by extending the forward-backward martingale decomposition of Lyons-Meyer-Zheng to the Hilbert space valued additive functionals associated with general non-reversible Markov processes.     

Déviations modérées pour la moyennisation d'une EDS avec petite diffusion

We prove in this Note the moderate deviation principle (MDP) for the averaging principle of a stochastic differential equation (SDE) in a fast random environment, modelized by an exponentially ergodic Markov process independent of the Wiener process driving the SDE. The main tools will be the method of Puhalskii for exponential tightness and a MDP for inhomogeneous functionals of Markov processes established in [5].     

Countably many simultaneous random changes of time scales

Simultaneous changes of time scales of the components of a vector Markov process are defined and developed. Measurability properties, Dynkin's lemma, and the strong Markov property are established for the transformed process.     

Entropy and the fourth moment phenomenon

We develop a new method for bounding the relative entropy of a random vector in terms of its Stein factors. Our approach is based on a novel representation for the score function of smoothly perturbed random variables, as well as on the de Bruijn?s formula of information theory. When applied to sequences of functionals of a general Gaussian field, our results can be combined with the Carbery–Wright inequality in order to yield multidimensional entropic rates of convergence that coincide, up to a logarithmic factor, with those achievable in smooth distances (such as the 1-Wasserstein distance). In particular, our findings settle the open problem of proving a quantitative version of the multidimensional fourth moment theorem for random vectors having chaotic components, with explicit rates of convergence in total variation that are independent of the order of the associated Wiener chaoses. The results proved in the present paper are outside the scope of other existing techniques, such as for instance the multidimensional Stein?s method for normal approximations.     

A Markov Property for Set-Indexed Processes

We consider a type of Markov property for set-indexed processes which is satisfied by all processes with independent increments and which allows us to introduce a transition system theory leading to the construction of the process. A set-indexed generator is defined such that it completely characterizes the distribution of the process.     

Malliavin calculus, geometric mixing, and expansion of diffusion functionals

Under geometric mixing condition, we presented asymptotic expansion of the distribution of an additive functional of a Markov or an ε-Markov process with finite autoregression including Markov type semimartingales and time series models with discrete time parameter. The emphasis is put on the use of the Malliavin calculus in place of the conditional type Cramér condition, whose verification is in most case not easy for continuous time processes without such an infinite dimensional approach. In the second part, by means of the perturbation method and the operational calculus, we proved the geometric mixing property for non-symmetric diffusion processes, and presented a sufficient condition which is easily checked in practice. Accordingly, we obtained asymptotic expansion of diffusion functionals and proved the validity of it under mild conditions, e.g., without the strong contractivity condition. Received: 7 September 1997 / Revised version: 17 March 1999     

Feller Property and Infinitesimal Generator of the Exploration Process

We consider the exploration process associated to the continuous random tree (CRT) built using a Lévy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale. The research of the second author was partially supported by NSERC Discovery Grants of the Probability group at Univ. of British Columbia.     

On the first birth and the last death in a generation in a multi-type Markov branching process

In a multi-type continuous time Markov branching process the asymptotic distribution of the first birth in and the last death (extinction) of the kth generation can be determined from the asymptotic behavior of the probability generating function of the vector Z(^k)(t), the size of the kth generation at time t, as t tends to zero or as t tends to infinity, respectively. Apart from an appropriate transformation of the time scale, for a large initial population the generations emerge according to an independent sum of compound multi-dimensional Poisson processes and become extinct like a vector of independent reversed Poisson processes. In the first birth case the results also hold for a multi-type Bellman-Harris process if the life span distributions are differentiable at zero.     

On Supercontractivity for Markov Semigroups

In this paper, we study the supercontractivity for Maxkov semigroups and obtain some sufficient and necessary conditions; especially explicit formulae are obtained for birth-death process and diffusion on the line. Sufficient conditions and necessary conditions in terms of isoperimetric inequalities are also presented. Moreover, we prove that the supercontractivity is equivalent to the compact embedding of Sobolev space into an Orlicz space.     

Existence and stability for Fokker–Planck equations with log-concave reference measure

We study Markov processes associated with stochastic differential equations, whose non-linearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a Fokker–Planck equation as the steepest descent flow of the relative entropy functional in the space of probability measures, endowed with the Wasserstein distance.     

A time-continuous markov chain interest model with applications to insurance

The force of interest is modelled by a homogeneous time-continuous Markov chain with finite state space. Ordinary differential equations are obtained for expected values of various functionals of this process, in particular for moments of present values of payment streams that may be deterministic or, possibly, also stochastic and driven by a time-continuous Markov chain. The homogeneity of the interest process gives rise to explicit formulae for expected values of some stationary functionals, e.g. moments of a perpetuity. Applications are made to some standard forms of insurance.     

Tensor Approximation of Generalized Correlated Diffusions and Functional Copula Operators

In this paper we develop a class of applied probabilistic continuous time but discretized state space decompositions of the characterization of a multivariate generalized diffusion process. This decomposition is novel and, in particular, it allows one to construct families of mimicking classes of processes for such continuous state and continuous time diffusions in the form of a discrete state space but continuous time Markov chain representation. Furthermore, we present this novel decomposition and study its discretization properties from several perspectives. This class of decomposition both brings insight into understanding locally in the state space the induced dependence structures from the generalized diffusion process as well as admitting computationally efficient representations in order to evaluate functionals of generalized multivariate diffusion processes, which is based on a simple rank one tensor approximation of the exact representation. In particular, we investigate aspects of semimartingale decompositions, approximation and the martingale representation for multidimensional correlated Markov processes. A new interpretation of the dependence among processes is given using the martingale approach. We show that it is possible to represent, in both continuous and discrete space, that a multidimensional correlated generalized diffusion is a linear combination of processes originated from the decomposition of the starting multidimensional semimartingale. This result not only reconciles with the existing theory of diffusion approximations and decompositions, but defines the general representation of infinitesimal generators for both multidimensional generalized diffusions and, as we will demonstrate, also for the specification of copula density dependence structures. This new result provides immediate representation of the approximate weak solution for correlated stochastic differential equations. Finally, we demonstrate desirable convergence results for the proposed multidimensional semimartingales decomposition approximations.     

Moderate deviations for Markov chains with atom

We obtain in this paper moderate deviations for functional empirical processes of general state space valued Markov chains with atom under weak conditions: a tail condition on the first time of return to the atom, and usual conditions on the class of functions. Our proofs rely on the regeneration method and sharp conditions issued of moderate deviations of independent random variables. We prove our result in the nonseparable case for additive and unbounded functionals of Markov chains, extending the work of de Acosta and Chen (J. Theoret. Probab. (1998) 75–110) and Wu (Ann. Probab. (1995) 420–445). One may regard it as the analog for the Markov chains of the beautiful characterization of moderate deviations for i.i.d. case of Ledoux 1992. Some applications to Markov chains with a countable state space are considered.