Convergence,boundedness, and ergodicity of regime-switching diusion processes with infinite memory

We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t): Under suitable conditions, we investigate convergence and boundedness of both the solutions X(t) and the functional solutions X_t: We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process (X_t,Λ(t)); and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.     

Coupling and harmonic functions in the case of continuous time Markov processes

Consider two transient Markov processes (X^v_t)_tεR·, (X^μ_t)_tεR· with the same transition semigroup and initial distributions v and μ. The probability spaces supporting the processes each are also assumed to support an exponentially distributed random variable independent of the process.

We show that there exist (randomized) stopping times S for (X^v_t), T for (X^μ_t) with common final distribution, L(X^v_S|S < ∞) = L(X^μ_T|T < ∞), and the property that for t < S, resp. t < T, the processes move in disjoint portions of the state space. For such a coupling (S, T) it is shown

where denotes the bounded harmonic functions of the Markov transition semigroup. Extensions, consequences and applications of this result are discussed.     

A stochastic maximum principle for processes driven by fractional Brownian motion

We prove a stochastic maximum principle for controlled processes X(t)=X^(u)(t) of the form

dX(t)=b(t,X(t),u(t)) dt+σ(t,X(t),u(t)) dB^(H)(t),

where B^(H)(t) is m-dimensional fractional Brownian motion with Hurst parameter . As an application we solve a problem about minimal variance hedging in an incomplete market driven by fractional Brownian motion.     

Multifractal analysis of the occupation measures of a kind of stochastic processes

Let {X(t), 0t1} be a stochastic process whose range is a random Cantor-like set depending on an -sequence (0<<1) and μ is the occupation measure of X(t). In this paper we examine the multifractal structure of μ and obtain the fractal dimensions of the sets of points of where the local dimension of μ is different from . It is interesting to notice that the final results of this paper are identical to those for the occupation measure of a stable subordinator with index , yet the stochastic process under consideration in this work is not even a Markov process.     

单边合格品链长中位数控制图设计

针对过程数据存在异常值的问题,为了监控过程均值的偏移,采用中位数统计量(X)代替传统均值(X)统计量,提出了一种单边合格品链长X(Sided Sensitive Conforming Run Length X, SCRL & X)控制图。采用马尔科夫链方法研究了SCRL & X控制图的性能,首先推导出其一步状态转移矩阵,进一步根据马尔科夫链的性质得到其平均链长(Average Run Length, ARL)。为了获得控制图的最优设计参数和性能指标值,保证其处于过程受控状态下的性能,并使其处于过程失控状态下的平均链长最小。研究结果表明,提出的SCRL & X控制图的统计性能优于传统的双边合格品链长X(Conforming Run Length, CRL & X)控制图,尤其针对过程均值产生较小偏移的情形,其优势较为明显。     

A note on the stationarity of a threshold first-order bilinear process

In the present note we study the threshold first-order bilinear model

X(t)=aX(t−1)+(b₁1_{X(t−1)<c}+b₂1_{X(t−1)c})X(t−1)e(t−1)+e(t), tεN

where {e(t), tεN} is a sequence of i.i.d. absolutely continuous random variables, X(0) is a given random variable and a, b₁, b₂ and c are real numbers. Under suitable conditions on the coefficients and lower semicontinuity of the densities of the noise sequence, we provide sufficient conditions for the existence of a stationary solution process to the present model and of its finite moments of order p.     

The consistency of a non-linear least squares estimator from diffusion processes

Consider the following Itô stochastic differential equation dX(t) = ƒ(θ⁰, X(t)) dt + dW(t), where (W(t), t 0), is a standard Wiener process in R^N. On the basis of discrete data 0 = t₀ < t₁ < …<t_n = T; X(t₁),...,X(t_n) we would like to estimate the parameter θ⁰. We shall define the least squares estimator and show that under some regularity conditions, is strongly consistent.     

Persistent convergence on randomly deleted sets

At time t_k, a unit with magnitude X_k and lifetime L_k enters a system. Let λ be a real valued function on the finite real sequences. One such sequence, B*_t, consists of the X_k's for which t_k t < t_k + L_k. When λ(X₁,…, X_n) converges (in some sense) to φ, we find conditions under which λ(B*_t) converges or fails to converge to φ in the same sense.     

Computing simple paths among obstacles

Given a set X of points in the plane, two distinguished points s,tX, and a set Φ of obstacles represented by line segments, we wish to compute a simple polygonal path from s to t that uses only points in X as vertices and avoids the obstacles in Φ. We present two results: (1) we show that finding such simple paths among arbitrary obstacles is NP-complete, and (2) we give a polynomial-time algorithm that computes simple paths when the obstacles form a simple polygon P and X is inside P. Our algorithm runs in time O(m²n²), where m is the number of vertices of P and n is the number of points in X.     

On mixtures of distributions of Markov chains

Let X be a chain with discrete state space I, and V be the matrix of entries V_i,n, where V_i,n denotes the position of the process immediately after the nth visit to i. We prove that the law of X is a mixture of laws of Markov chains if and only if the distribution of V is invariant under finite permutations within rows (i.e., the V_i,n's are partially exchangeable in the sense of de Finetti). We also prove that an analogous statement holds true for mixtures of laws of Markov chains with a general state space and atomic kernels. Going back to the discrete case, we analyze the relationships between partial exchangeability of V and Markov exchangeability in the sense of Diaconis and Freedman. The main statement is that the former is stronger than the latter, but the two are equivalent under the assumption of recurrence. Combination of this equivalence with the aforesaid representation theorem gives the Diaconis and Freedman basic result for mixtures of Markov chains.     

A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model

For the pth-order linear ARCH model,

, where ₀ > 0, _i 0, I = 1, 2, …, p, {_t} is an i.i.d. normal white noise with E_t = 0, E_t² = 1, and _t is independent of {X_s, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, ₁ + ₂ + ··· + _p < 1. In this note, we assume that _t has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under E_t² = 1. When _t has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.     

Siegmund Duality for Continuous Time Markov Chains on ℤ<Stack><Subscript>+</Subscript><Superscript>d</Superscript></Stack>

For the continuous time Markov chain with transition function P(t) on Z₊^d, we give the necessary and sufficient conditions for the existence of its Siegmund dual with transition function P(t). If Q, the q-matrix of P(t), is uniformly bounded, we show that the Siegmund dual relation can be expressed directly in terms of q-matrices, and a sufficient condition under which the Q-function is the Siegmund dual of some Q-function is also given.     

The problem of minimax mean square filtration in parabolic systems : PMM vol. 42, no. 6, 1978, pp, 1016–1025

The problem of estimating parameters of state of a distributed parabolic system by observation results is considered. The system is assumed to function under conditions of undefined perturbations in the measurement channel and specified initial distribution. The problem is considered in minimax formulation [1] in conformity with the scheme accepted for ordinary differential equations [2].(^*), Analytic definition of sets X (/gJ, y (·)) (/gJ > 0) of states of a parabolic system compatible at instant /gJ with the realizable signal y (t) (t ε [0, /gJ]) is obtained. An element of region X (/gJ, y (·)) which satisfies the specified minimax criterion is chosen as the optimal estimate of the true state at instant /gJ. Integradifferential equations in partial derivatives are derived for parameters that define the evolution of regions X (/gJ, y (·)) in time. One of the methods of approximating the input problem of observation by similar problems for systems of ordinary differential equations is discussed on a specific example. Problems of observation for distributed systems in different formulations appear in [3 – 6].     

An asymptotic representation for products of random matrices

Let {X_k} be a stationary ergodic sequence of nonnegative matrices. It is shown in this paper that, under mild additional conditions, the logarithm of the i, jth element of X_t···X₁ is well approximated by a sum of t random variables from a stationary ergodic sequence. This representation is very useful for the study of limit behaviour of products of random matrices. An iterated logarithm result and an estimation result of use in the theory of demographic population projections are derived as corollaries.     

Caractérisation spectrale de la forme de Jordan

Soit , où désigne l'ensemble des matrices n×n à coefficients complexes. Nous montrons qu'on peut complètement caractériser la forme de Jordan de A en examinant le polynôme caractéristique de tA+X pour tous les tC et tous les . Ceci nous permet de donner une démonstration plus élémentaire d'un théorème de Baribeau et Ransford sur les transformations holomorphes de qui préservent le spectre.

Denote by the set of complex n×n matrices, and let . We give a variational, purely spectral characterization of the Jordan form of A by examining the characteristic polynomial of the perturbed matrices tA+X for tC and . This allows us to give a more elementary proof of a theorem of Baribeau and Ransford on spectrum-preserving holomorphic maps on .     

Function-space compactifications of function spaces

If X and Y are Hausdorff spaces with X locally compact, then the compact-open topology on the set C(X,Y) of continuous maps from X to Y is known to produce the right function-space topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for any Tychonoff space Y, there is a densely injective space Z containing Y as a densely embedded subspace such that, for every locally compact space X, the set C(X,Z) has a compact Hausdorff topology whose relative topology on C(X,Y) is the compact-open topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X,Y) under the compact-open topology is embedded into the Vietoris hyperspace V(X×Y). (2) The space of real-valued continuous functions on a locally compact Hausdorff space under the compact-open topology is embedded into a compact Hausdorff space whose points are pairs of extended real-valued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways.     

Additive Maps Preserving Nilpotent Perturbation of Scalars

Let X be a Banach space over F(= R or C) with dimension greater than 2. Let N(X) be the set of all nilpotent operators and B_0(X) the set spanned by N(X). We give a structure result to the additive maps on FI + B_0(X) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on FI + B_0(X) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T) = cAT A~(-1)+ φ(T)I for all T ∈ FI + B_0(X) or Φ(T) = cAT*A~(-1)+ φ(T)I for all T ∈ FI + B_0(X), where c is a nonzero scalar,A is a τ-linear bijective transformation for some automorphism τ of F and φ is an additive functional.In addition, if dim X = ∞, then A is in fact a linear or conjugate linear invertible bounded operator.     

Neighborhood assignments and cardinal functions: a unified approach to metrization and uniformity

We use neighborhood assignments and cardinal functions to give a unified approach to metrizability and uniformity. This leads to a number of characterizations of m(X), the metrizability degree of X, u(X), the uniform weight of X, and w(X), the weight of X. For X normal (and regular), m(X) = u(X); it is unknown whether this result extends to completely regular spaces.     

Coincidence of the upper Kuratowski topology with the co-compact topology on compact sets, and the Prohorov property

For a Hausdorff space X, let F be the hyperspace of all closed subsets of X and H a sublattice of F. Following Nogura and Shakhmatov, X is said to be H-trivial if the upper Kuratowski topology and the co-compact topology coincide on H. F-trivial spaces are the consonant spaces first introduced and studied by Dolecki, Greco and Lechicki. In this paper, we deal with K-trivial spaces and Fin-trivial space, where K and Fin are respectively the lattices of compact and of finite subsets of X. It is proved that if C_k(X) is a Baire space or more generally if X has ‘the moving off property’ of Gruenhage and Ma, then X is K-trivial. If X is countable, then C_p(X) is Baire if and only if X is Fin-trivial and all compact subsets of X are finite. As for consonant spaces, it turns out that every regular K-trivial space is a Prohorov space. This result remains true for any regular Fin-trivial space in which all compact subsets are scattered. It follows that every regular first countable space without isolated points, all compact subsets of which are countable, is Fin-nontrivial. Examples of K-trivial non-consonant spaces, of Fin-trivial K-nontrivial spaces and of countably compact Prohorov Fin-nontrivial spaces, are given. In particular, we show that all (generalized) Fréchet–Urysohn fans are K-trivial, answering a question by Nogura and Shakhmatov. Finally, we describe an example of a continuous open compact-covering mapping f :X→Y, where X is Prohorov and Y is not Prohorov, answering a long-standing question by Topsøe.     

Multivariate normality via conditional specification

If X is a k-dimensional random vector, we denote by X_(i) the vector X with coordinate i deleted and by X_(i,j) the vector X with coordinates i and j deleted. If for each i the conditional distribution of X_i given X_(i) = x_(i) is univariate normal for each x_(i) ^K−1 and if for each i, j the conditional distribution of X_i given X_(i,j) = x_(i,j) is univariate normal for each x_(i,j) ^k−2 then it is shown that X has a classical k-variate normal distribution.