Excess-time Renewal Theory with Applications

In this paper, the renewal theory is investigated from the point of view of the excess-time (non-excess-time) renewal intervals. Useful results are derived, and applications to several renewal models are given.     

Discrete Periodic Sampling with Jitter and Almost Periodically Correlated Processes

The zero-mean process is said to be almost periodically correlated whenever its shifted covariance kernel is almost periodic in t uniformly with respect to . Then it admits a Fourier–Bohr decomposition: . This paper deals with the estimation of the spectral covariance a(λ,τ) from a discrete time observation of the process , when jitter and delay phenomena are present in conjunction with periodic sampling. Under mixing conditions, we establish the consistency and the asymptotic normality of empirical estimators as the sampling time step tends to 0 and the sampling period tends to infinity.      

On Non-Continuous Dirichlet Processes

We introduce here some Itô calculus for non-continuous Dirichlet processes. Such calculus extends what was known for continuous Dirichlet processes or for semimartingales. In particular we prove that non-continuous Dirichlet processes are stable under C ¹ transformation.     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

   Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par     

On Markov-Additive Jump Processes

In 1995, Pacheco and Prabhu introduced the class of so-called Markov-additive processes of arrivals in order to provide a general class of arrival processes for queueing theory. In this paper, the above class is generalized considerably, including time-inhomogeneous arrival rates, general phase spaces and the arrival space being a general vector space (instead of the finite-dimensional Euclidean space). Furthermore, the class of Markov-additive jump processes introduced in the present paper is embedded into the existing theory of jump processes. The best known special case is the class of BMAP arrival processes.     

On Fractional Brownian Processes

We use Liouville spaces in order to prove the existence of some different fractional -Brownian motion ( 0 < 1 ), or fractional ( , )-Brownian sheets. There are also applications to the Wiener stochastic integral with respect to these -Brownian.     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par     

On the Approximation of Quantile Processes by Kiefer Processes

We prove that the best possible almost sure rate of uniform approximation of a uniform quantile process by a normed Kiefer process is O(n ^–1/4(log n)^1/2× (log log n)^1/4).     

Pricing Vulnerable Options with Correlated Credit Risk Under Jump-diffusion Processes When Corporate Liabilities Are Random

Acta Mathematicae Applicatae Sinica, English Series - In this paper, we consider an improved model of pricing vulnerable options with credit risk. We assume that the vulnerable European options not...     

On Non-simple Marked Point Processes

Simple point processes are often characterized by their associated compensators or conditional intensities. Non-simple point processes are not uniquely determined by their conditional intensity and compensator, so instead one may identify with the point process its associated simple point process and corresponding conditional intensity, on an expanded mark space. Some relations between the conditional intensity on the expanded mark space and the ordinary conditional intensity are investigated here, and some classes of separable non-simple processes are presented. Transformations into simple point processes, involving thinning and rescaling, are presented.     

On Sampling Stationary Stochastic Processes

A quasi Fourier-type duality associated with a bandlimited stationary stochastic process can be established. It comes from the spectral representation of the process and a compact support assumption for its spectral density. In this way, for essentially bounded spectral densities we have an isometry between a weightedL²space and the Hilbert space spanned by the process. We can transfer converging expansions for the exponential complex e^itw in the :L²-space into a sampling expansion for the process converging in the mean square sense.     

On Multidimensional Generalized Diffusion Processes

We construct a multidimensional generalized diffusion process with the drift coefficient that is the (generalized) derivative of a vector-valued measure satisfying an analog of the Hölder condition with respect to volume. We prove the existence and continuity of the density of transition probability of this process and obtain standard estimates for this density. We also prove that the trajectories of the process are solutions of a stochastic differential equation.     

On weakly mixing Markov Processes

A result of England and Martin on weak mixing (see [6]) is extended to Markov Processes in a strengthened form, and also to continuous time Markov Processes.     

纯生过程的变异性(英语)

设{X(t):t≥0}为零初值纯生过程,出生率为λ_n,n≥0.在本文中,我们证明了Faddy[7]的一个猜测:当出生率为单调增加序列λ_0≤λ_1≤λ_2…。时,Var{X(t)}≥E{(t)};当出生率为单调减少序列时Var{X(t)}≤E{(t)}。     

ϡ�������һ�����������ַ���ģ�͵��Ʋ�����

??The paper considers a risk model with two dependent classes of
insurance business. In this model, the two claim number processes are partly sparsely
correlated through an Erlang(2) process. By introducing an auxiliary model, we obtain the
integral equations for ultimate ruin probabilities, and discuss the asymptotic property of
ruin probabilities by renewal approach. We also get the linear differential equations of
ruin probabilities of the model and the corresponding auxiliary model when claims follow
the exponential distributions, and show how solves the linear differential equations by a
specific example.     

On a Replacement Problem of a Cumulative Damage Model

Items are assumed to fail only by degradation. An appropriate stochastic model of such items is a cumulative process in which an item can fail only when the total amount of damage exceeds a prespecified failure level. This paper introduces a replacement policy in which an item is replaced at a certain level of damage before failure or at failure, whichever occurs first. The optimum replacement level of damage which will minimize the total expected cost per unit of time for an infinite time span is obtained. A numerical example is also presented. The total expected cost for a finite time span is also discussed.     

On Realizations from Nonstationary Time Processes

Unthinking inferences from the sampled autocorrelation structure, for realisations of certain nonstationary processes, can lead to misguided identifications and, possibly, misleading fits. Effective time-series modelling must include inspection of the raw data, and will often also depend on extraneous information available to the analyst.     

On Martingale Approximation of Adapted Processes

We show that the existence of a martingale approximation of a stationary process depends on the choice of the filtration. There exists a stationary linear process which has a martingale approximation with respect to the natural filtration, but no approximation with respect to a larger filtration with respect to which it is adapted and regular. There exists a stationary process adapted, regular, and having a martingale approximation with respect to a given filtration but not (regular and having a martingale approximation) with respect to the natural filtration.