关于Gauss过程增量的若干结果

Let {X(t);t≥0}be a Gaussian process with stationary increments,X(0)=0(a.s.),EX(t)=0 andσ~2(h)=EX(t+h)-X(t)~=EZ~2(h)=Coh~(2α),0<α≤1.In this paper,we first prove that the Levy's theorem of the modulus of continuityof the Wiener process is also true for {X(t);t≥0};i.e.Furthermore,we point out that some results on increments of the Wiener processesin[3]and[4]remain true for the increments of {X(t);t≥0}.     

Ruin Probabilities under a Markovian Risk Model

In this paper, a Markovian risk model is developed, in which the occurrence of the claims is described by a point process {N(t)}t≥0 with N(t) being the number of jumps of a Markov chain during the interval [0, t]. For the model, the explicit form of the ruin probability ψ(0) and the bound for the convergence rate of the ruin probability ψ(u) are given by using the generalized renewal technique developed in this paper.Finally, we prove that the ruin probability ψ(u) is a linear combination of some negative exponential functions in a special case when the claims are exponentially distributed and the Markov chain has an intensity matrix(qij)i,j∈E such that qm = qml and qi=qi(i 1), 1≤i≤m-1.     

PARAMETER ESTIMATION FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A GENERAL GAUSSIAN NOISE

In this paper,we consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process(G_t)t≥0.The second order mixed partial derivative of the covariance function R(t,s)=E[GtGs]can be decomposed into two parts,one of which coincides with that of fractional Brownian motion and the other of which is bounded by(ts)^β-1up to a constant factor.This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or to have stationary increments;some examples of this include the subfractional Brownian motion and the bi-fractional Brownian motion.Under this assumption,we study the parameter estimation for a drift parameter in the Ornstein-Uhlenbeck process driven by the Gaussian noise(G_t)t≥0.For the least squares estimator and the second moment estimator constructed from the continuous observations,we prove the strong consistency and the asympotic normality,and obtain the Berry-Esséen bounds.The proof is based on the inner product's representation of the Hilbert space(h)associated with the Gaussian noise(G_t)t≥0,and the estimation of the inner product based on the results of the Hilbert space associated with the fractional Brownian motion.     

A branching particle system approximation for nonlinear stochastic filtering

The optimal filter π = {π t,t ∈ [0,T ]} of a stochastic signal is approximated by a sequence {π n t } of measure-valued processes defined by branching particle systems in a random environment(given by the observation process).The location and weight of each particle are governed by stochastic differential equations driven by the observation process,which is common for all particles,as well as by an individual Brownian motion,which applies to this specific particle only.The branching mechanism of each particle depends on the observation process and the path of this particle itself during its short lifetime δ = n 2α,where n is the number of initial particles and α is a fixed parameter to be optimized.As n →∞,we prove the convergence of π n t to π t uniformly for t ∈ [0,T ].Compared with the available results in the literature,the main contribution of this article is that the approximation is free of any stochastic integral which makes the numerical implementation readily available.     

QUASI-STATIONARY DISTRIBUTIONS FOR THE RADIAL ORNSTEIN-UHLENBECK PROCESSES

The purpose of this article is to obtain the quasi-stationary distributions of the δ（δ 〈 2）-dimensional radial Ornstein-Uhlenbeck process with parameter -λ by using the methods of Martinez and San Martin （2001）. It is described that the law of this process conditioned on first hitting 0 is just the probability measure induced by a （4 - δ）- dimensional radial Ornstein-Uhlenbeck process with parameter -λ. Moreover, it is shown that the law of the conditioned process associated with the left eigenfunction of the process conditioned on first hitting 0 is induced by a one-parameter diffusion.     

DECOMPOSITION OF A CLASS OF FUNCTIONALS AND THE PREDICTABLE REPRESENTATION THEOREM ON BANACH SPACES

Let B be a separable real Banach space and X(t) be a symmetric conservative diffusionprocess taking values in B. In this paper, we decompose the functional u(X(t),t) into a sumof a square integrable martingale and a regular 0-quadratic variation process. On this basis, weestablish the predictable representation theorem of X(t).     

LEAST SQUARES ESTIMATION FOR ORNSTEIN-UHLENBECK PROCESSES DRIVEN BY THE WEIGHTED FRACTIONAL BROWNIAN MOTION

In this article, we study a least squares estimator(LSE) of θ for the OrnsteinUhlenbeck process X_0=0, dX_t =θX_tdt + dB_t~(a,b), t≥ 0 driven by weighted fractional Brownian motion B~(a,b) with parameters a, b. We obtain the consistency and the asymptotic distribution of the LSE based on the observation {X_s, s ∈ [0, t]} as t tends to infinity.     

A NOTE ON THE CHARACTERISTICS OF POISSON PROCESSES

In this note two characteristic theorems of Poisson processes are given.If{N(t);t≥0}is arenewal process,U_t,V_t are,respectively,the time since the last renewal and the time to the nextrenewal at t,Z_t=U_t+V_t,then a Poisson process can be characterized by the limiting independenceof the joint distribution of(U_t,V_t)when t→∞,or EZ_t,or the distribution of Z_t.     

STRONG APPROXIMATION FOR MOVING AVERAGE PROCESSES UNDER DEPENDENCE ASSUMPTIONS

Let {Xt,t ≥ 1} be a moving average process defined by Xt = ∑^∞ k=0 αkξt-k, where {αk,k ≥ 0} is a sequence of real numbers and {ξt,-∞ 〈 t 〈 ∞} is a doubly infinite sequence of strictly stationary dependent random variables. Under the conditions of {αk, k ≥ 0} which entail that {Xt, t ≥ 1} is either a long memory process or a linear process, the strong approximation of {Xt, t ≥ 1} to a Gaussian process is studied. Finally, the results are applied to obtain the strong approximation of a long memory process to a fractional Brownian motion and the laws of the iterated logarithm for moving average processes.     

Some Limit Theorems in Geometric Processes

Geometric process (GP) was introduced by Lam[4,5], it is defined as a stochastic process {Xn, n = 1, 2,…} for which there exists a real number a > 0, such that {an-1 Xn, n = 1,2, …} forms a renewal process (RP). In this paper, we study some limit theorems in GP. We first derive the Wald equation for GP and then obtain the limit theorems of the age, residual life and the total life at t for a GP. A general limit theorem for Sn with a > 1 is also studied. Furthermore, we make a comparison between GP and RP, including the comparison of their limit distributions of the age, residual life and the total life at t.     

Precise rate in the law of iterated logarithm for <Emphasis Type="Italic">ρ</Emphasis>-mixing sequence

Let {X, X _n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set . Suppose lim _n→∞ and , where d=2, if −1<b<0 and d>2(b+1), if b≥0. It is proved that, for any b>−1,

具时滞的奇异(n-1,1)共轭边值问题的多重正解

Abstract. This paper discusses the singular (n-l, 1) conjugate boundary value problem as fol-lows by using a fixed point index theorem in cones     

Existence of Positive Solutions for a Class of Sigular Boundary Value Problem of Fourth Order

Some results concerning existence of positive solutions for the singular boundary value problems u^(4)(t)=f(t,u(t)) t∈（0,1) u(0)=u(1)=0 u‘(0)=u‘(1)=0 have been given, where f(t, x) may be singular at t = 0, 1.     

Sufficient and necessary conditions for limit laws on lag sums of I. I. D. R. V. S

Let {X, X _n ;n>-1} be a sequence of i.i.d.r.v.s withEX=0 andEX ²=σ²(0 < σ < ∞). we obtain some sufficient and necessary conditions for

On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

Oscillation results for second order nonlinear differential equations

In this paper, the authors present some new results for the oscillation of the second order nonlinear neutral differential equations of the form

Existence of type (II) regions and convexity and concavity of potential functionals corresponding to jumping nonlinear problems

In this paper we study the jumping nonlinear problem

The existence of multiple positive solutions of p-Laplacian boundary value problems

In this paper, we establish sufficient conditions to guarantee the existence of at least three or 2n − 1 positive solutions of nonlocal boundary value problems consisting of the second-order differential equation with p-Laplacian