Two-Parameter Lévy Processes Along Decreasing Paths

Let {X_t1,t2:t₁,t₂ 3 0}\{X_{t_{1},t_{2}}:t_{1},t_{2}\geq0\} be a two-parameter Lévy process on ℝ ^d . We study basic properties of the one-parameter process {X _x(t),y(t):t∈T} where x and y are, respectively, nondecreasing and nonincreasing nonnegative continuous functions on the interval T. We focus on and characterize the case where the process has stationary increments.     

On–off fluid models in heavy traffic environment

We consider fluid models with infinite buffer size. Let {Z _N (t)} be the net input rate to the buffer, where {{Z _N (t)} is a superposition of N homogeneous alternating on–off flows. Under heavy traffic environment {{Z _N (t)} converges in distribution to a centred Gaussian process with covariance function of a single flow. The aim of this paper is to prove the convergence of the stationary buffer content process {X _N ^* (t)} in the fNth model to the buffer content process {X _N (t)} in the limiting Gaussian model. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Generalized Multifractional Field: A Nice Tool for the Study of the Generalized Multifractional Brownian Motion

The Generalized Multifractional Brownian Motion (GMBM) is a continuous Gaussian process {X(t)}_{t ? [0,1]}\{X(t)\}_{t\in [0,1]} that extends the classical Fractional Brownian Motion (FBM) and the Multifractional Brownian Motion (MBM) [15, 4, 1, 1]. Its main interest is that, its Hölder regularity can change widely from point to point. In this article we introduce the Generalized Multifractional Field (GMF), a continuous Gaussian field {Y(x,y)}_{(x,y) ? [0,1] ²}\{Y(x,y)\}_{(x,y)\in [0,1]^{\,2}} that satisfies for every tt, X(t)=Y(t,t)X(t)=Y(t,t). Then, we give a wavelet decomposition of YY and using this nice decomposition, we show that YY is b\beta-Hölder in yy, uniformly in xx. Generally speaking this result seems to be quite important for the study of the GMBM. In this article, it will allow us to determine, without any restriction, its pointwise, almost sure, Hölder exponent and to prove that two GMBM's with the same Hölder regularity differ by a "smoother' process.     

On solutions of stochastic differential equations with drift

Summary We study stochastic differential equations of the formdX _t=(X _t)dM_t+b(X_t)d_t whereM is a continuous local martingale and <M> stands for its quadratic variation process. The conditions introduced by Engelbert and Schmidt, which ensure the existence and uniqueness in law of solutions of SDE's driven by the Wiener process without drift (or with generalized drift) are shown to be no longer valid.     

Some ratio inequalities for iterated stochastic integrals

Let X = (X_t, ?_t) be a continuous local martingale with quadratic variation 〈X〉 and X₀ = 0. Define iterated stochastic integrals I_n(X) = (I_n(t, X), ?_t), n ≥ 0, inductively by $$ I_{n} (t, X) = \int ^{t} _{0} I_{n-1} (s, X)dX_{s} $$ with I₀(t, X) = 1 and I₁(t, X) = X_t. Let (??^x_t(X)) be the local time of a continuous local martingale X at x ∈ ?. Denote ??*t(X) = sup_x∈? ??^x_t(X) and X* = sup_t≥0 |X_t|. In this paper, we shall establish various ratio inequalities for I_n(X). In particular, we show that the inequalities $$ c_{n,p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} \; \le \; \left\Vert {\mathop \sup \limits _{t \ge 0}} \; {\left\vert I_{n} (t, X) \right\vert \over {(1+ \langle X \rangle _{t} ) ^{n/2}}} \right\Vert _{p} \; \le C_{n, p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} $$ hold for 0 < p < ∞ with some positive constants c_n,p and C_n,p depending only on n and p, where G(t) = log(1+ log(1+ t)). Furthermore, we also show that for some γ ≥ 0 the inequality $$ E \left[ U ^{p}_{n} \exp \left( \gamma {U ^{1/n} _{n} \over {V}} \right) \right] \le C_{n, p, \gamma} E [V ^{n, p}] \quad (0 < p < \infty ) $$ holds with some positive constant C_n,p,γ depending only on n, p and γ, where U_n is one of 〈I_n(X)〉^1/2_∞ and I*_n(X), and V one of the three random variables X*, 〈X〉^1/2_∞ and ??*_∞(X). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Set-Indexed Conditional Empirical and Quantile Processes Based on Dependent Data

We consider a conditional empirical distribution of the form F_n(C x)=∑ⁿ_t=1 ω_n(X_t−x) I{Y_tC} indexed by C , where {(X_t, Y_t), t=1, …, n} are observations from a strictly stationary and strong mixing stochastic process, {ω_n(X_t−x)} are kernel weights, and is a class of sets. Under the assumption on the richness of the index class in terms of metric entropy with bracketing, we have established uniform convergence and asymptotic normality for F_n(· x). The key result specifies rates of convergences for the modulus of continuity of the conditional empirical process. The results are then applied to derive Bahadur–Kiefer type approximations for a generalized conditional quantile process which, in the case with independent observations, generalizes and improves earlier results. Potential applications in the areas of estimating level sets and testing for unimodality (or multimodality) of conditional distributions are discussed.     

Superposition of Spatially Interacting Aggregated Continuous Time Markov Chains

A system s{ X(t)} = {X ₁(t),X ₂(t),..., X _N(t)} of N interacting time reversible continuous time Markov chains is considered. The state space of each of the processes {X _i(t)} (i = 1, 2,...,N) is partitioned into two aggregates. Interaction between the processes {X _i(t)},{X ₂(t)},...,{X _N(t)} is introduced by allowing the transition rates of an individual process at time t to depend on the configuration of aggregates occupied by the other N - 1 processes at that time. The motivation for this work comes from ion channel modeling, where {(X}(t)} describes the gating mechanisms of N channels and the partitioning of the state space of {X _i(t)} correspond to whether the channel is conducting or not. Let S(t) denote the number of conducting channels at time t. For a time-reversible class of such processes, expressions are derived for the mean and probability density function of the sojourns of {S(t)} at its different levels when {X(t)} is in equilibrium. Particular attention is paid to the situation when the N channels are located on a circle with nearest neighbor interaction. Necessary and sufficient conditions for a general co-operative multiple channel system to be time reversible are derived.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA     

Hitting and martingale characterizations of one-dimensional diffusions

The main theorem of the paper is that, for a large class of one-dimensional diffusions (i. e. strong Markov processes with continuous sample paths): if x(t) is a continuous stochastic process possessing the hitting probabilities and mean exit times of the given diffusion, then x(t) is Markovian, with the transition probabilities of the diffusion. For a diffusion x(t) with natural boundaries at ± ∞, there is constructed a sequence π _n (t, x) of functions with the property that the π _n (t, x (t)) are martingales, reducing in the case of the Brownian motion to the familiar martingale polynomials. It is finally shown that if a stochastic process x (t) is a martingale with continuous paths, with the additional property that

$$\mathop \smallint \limits_0^{x\left( t \right)} m\left( {0,y} \right]dy - t$$     

Weak Solutions to Stochastic Wave Equations with Values in Riemannian Manifolds

Let M be a compact Riemannian manifold. We prove existence of a global weak solution of the stochastic wave equation D _t u _t  = D _x u _x  + (X _u  + λ₀(u)u _t  + λ₁(u)u _x )[Wdot] where X is a continuous vector field on M, λ₀ and λ₁ are continuous vector bundles homomorphisms from TM to TM, and W is a spatially homogeneous Wiener process on ? with finite spectral measure. We use recently introduced general method of constructing weak solutions of SPDEs that does not rely on any martingale representation theorem.     

Self-similar processes with independent increments

Summary A stochastic process {X _t t 0} onR ^d is called wide-sense self-similar if, for eachc>0, there are a positive numbera and a functionb(t) such that {X _ct } and {aX _t +b(t)} have common finite-dimensional distributions. If {X _t } is widesense self-similar with independent increments, stochastically continuous, andX ₀=const, then, for everyt, the distribution ofX _t is of classL. Conversely, if is a distribution of classL, then, for everyH>0, there is a unique process {X _(H) ^t } selfsimilar with exponentH with independent increments such thatX ₁ has distribution . Consequences of this characterization are discussed. The properties (finitedimensional distributions, behaviors for small time, etc.) of the process {X _(H) ^t } (called the process of classL with exponentH induced by ) are compared with those of the Lévy process {Y _t } such thatY ₁ has distribution . Results are generalized to operator-self-similar processes and distributions of classOL. A process {X _t } onR ^d is called wide-sense operator-self-similar if, for eachc>0, there are a linear operatorA _c and a functionb _c (t) such that {X _ct } and {A _c X _t +b _c (t)} have common finite-dimensional distributions. It is proved that, if {X _t } is wide-sense operator-self-similar and stochastically continuous, then theA _c can be chosen asA _c =c ^Q with a linear operatorQ with some special spectral properties. This is an extension of a theorem of Hudson and Mason [4].     

Some central limit theorems for ℓ∞-valued semimartingales and their applications

Summary. This paper is devoted to the generalization of central limit theorems for empirical processes to several types of ℓ^∞(Ψ)-valued continuous-time stochastic processes t⇝X _t ⁿ =(X _t ^{n ,ψ}|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X _t ^{n ,ψ} is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived. Received: 6 May 1996 / In revised form: 4 February 1997     

Self-interacting diffusions

 This paper is concerned with a general class of self-interacting diffusions {X _t } _{t ≥0} living on a compact Riemannian manifold M. These are solutions to stochastic differential equations of the form : dX _t = Brownian increments + drift term depending on X _t and μ _t , the normalized occupation measure of the process. It is proved that the asymptotic behavior of {μ _t } can be precisely related to the asymptotic behavior of a deterministic dynamical semi-flow Φ = {Φ _t } _{t ≥0} defined on the space of the Borel probability measures on M. In particular, the limit sets of {μ _t } are proved to be almost surely attractor free sets for Φ. These results are applied to several examples of self-attracting/repelling diffusions on the n-sphere. For instance, in the case of self-attracting diffusions, our results apply to prove that {μ _t } can either converge toward the normalized Riemannian measure, or to a gaussian measure, depending on the value of a parameter measuring the strength of the attraction. Received: 21 July 2000 / Revised version: 12 December 2000 / Published online: 15 October 2001     

Convergence in probability and almost sure with applications

A necessary and sufficient condition is given for the convergence in probability of a stochastic process {X_t}. Moreover, as a byproduct, an almost sure convergent stochastic process {Y_t} with the same limit as {X_t} is identified. In a number of cases {Y_t} reduces to {X_t} thereby proving a.s. convergence. In other cases it leads to a different sequence but, under further assumptions, it may be shown that {X_t} and {Y_t} are a.s. equivalent, implying that {X_t} is a.s. convergent. The method applies to a number of old and new cases of branching processes providing an unified approach. New results are derived for supercritical branching random walks and multitype branching processes in varying environment.     

On limit distributions of first crossing points of Gaussian sequences

Let {X_k, k?Z} be a stationary Gaussian sequence with EX₁ – 0, EX²_k = 1 and EX₀X_k = r_k. Define τ_x = inf{k: X_k >– βk} the first crossing point of the Gaussian sequence with the function – βt (β > 0). We consider limit distributions of τ_x as β→0, depending on the correlation function r_k. We generalize the results for crossing points τ_x = inf{k: X_k >β?(k)} with ?(– t)?t^γL(t) for t→∞, where γ > 0 and L(t) varies slowly.     

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.     

Stochastic integral representation of some martingales

Let {X_t} be a continuous square integrable martingale. Denote its increasing (natural) process by {A_t}. Let S_t, T_t be the left and right inverses of A_t, respectively. Then for any square integrable martingale {Y_t} defined on {X_t}, Y_t = ∝₀^tψ_sdX_s, R₀ < t < S_∞ where S_∞ = lim_t→∞S_t, R₀ = inf {t: X_t ≠ 0} provided that Y(T(t)) is σ(X(T(s)): s ? t)-measurable. All martingales are assumed to be zero at t = 0. Brownian motion and Poisson processes are considered also.     

Wellposedness of abstract Cauchy problems for second order differential equations

This paper concerns the abstract Cauchy problem (ACP) for an evolution equation of second order in time. LetA be a closed linear operator with domainD(A) dense in a Banach spaceX. We first characterize the exponential wellposedness of ACP onD(A ^k+1),k teN. Next let {C(t);t teR} be a family of generalized solution operators, on [D(A ^k)] toX, associated with an exponentially wellposed ACP onD(A ^k+1). Then we define a new family {T(t); Ret>0} by the abstract Weierstrass formula. We show that {T(t)} forms a holomorphic semigroup of class (H _k) onX. Research of the second-named author was partially supported by Grant-in-Aid for Scientific Research (No. 63540139), Ministry of Education, Science and Culture.     

Some path properties of generalized Lévy sheet

Let X(t) be an N parameter generalized Lévy sheet taking values in ℝ^d with a lower index α, ℜ = {(s, t] = ∏ _i=1 ^N (s _i, t _i], s _i < t _i}, E(x, Q) = {t ∈ Q: X(t) = x}, Q ∈ ℜ be the level set of X at x and X(Q) = {x: ∃t ∈ Q such that X(t) = x} be the image of X on Q. In this paper, the problems of the existence and increment size of the local times for X(t) are studied. In addition, the Hausdorff dimension of E(x, Q) and the upper bound of a uniform dimension for X(Q) are also established.