Smoothness of Gaussian local times beyond the local nondeterminism

The joint continuity of Gaussian local times is investigated under conditions strictly weaker than the local nondeterminism. Our conditions are given in terms of the interpolation variances only and they cover the class of Gaussian Markov processes. A new order of infinitesimal in the tail probability of the local time at the origin is obtained.     

On the most visited sites by a symmetric stable process

Summary. At time t, the most visited site of a linear Brownian motion is defined as the point which realises the supremum of the local times at time t. Let V be the time indexed process of the most visited sites by a linear Brownian motion. We show that every value is polar for V. Those results are extended from Brownian motion to symmetric stable processes, and then to the absolute value of a symmetric stable process. Received: 1 March 1996 / In revised form: 17 October 1996     

A canonical setting and separating times for continuous local martingales

The notion of a separating time for a pair of measures on a filtered space is helpful for studying problems of (local) absolute continuity and singularity of measures. In this paper, we describe a certain canonical setting for continuous local martingales (abbreviated below as CLMs) and find an explicit form of separating times for CLMs in this setting.     

On the density of the supremum of a stable process

We study the density of the supremum of a strictly stable Lévy process. Our first goal is to investigate convergence properties of the series representation for this density, which was established recently by Hubalek and Kuznetsov (2011) [24]. Our second goal is to investigate in more detail the important case when α$\alpha$ is rational: we derive an explicit formula for the Mellin transform of the supremum. We perform several numerical experiments and discuss their implications. Finally, we state some interesting connections that this problem has to other areas of Mathematics and Mathematical Physics and we also suggest several open problems.     

Forcing Divergence When the Supremum Is Not Integrable

If a sequence of functions (f_n) has an integrable supremum and converges almost everywhere, then an operator sequence (T_n) will yield a sequence (T_nf_n) that converges almost everywhere too, under some very general assumptions about the sequence (T_n). However, if the supremum of (f_n) is not integrable then this can fail to be the case. It is shown that if a sequence of functions (f_n)has a supremum sup_n |f_n| that is not integrable, then one can always construct a variety of sequences of positive contractions (T_n) such that lim sup |T_n f_n| = ∞ a.e. These operators can be conditional expectations with respect to an increasing sequence of finite σ-algebras, the conditional expectation with respect to one fixed σ-algebra, or averages with respect to a measure-preserving transformation. General discussion of these constructions, history of previous results of this type, and some open questions are also given.     

Intersection local times for interlacements

We define renormalized intersection local times for random interlacements of Lévy processes in R^d$R^d$ and prove an isomorphism theorem relating renormalized intersection local times with associated Wick polynomials.     

The supremum of Brownian local times on Hölder curves

For , we consider L^f_t, the local time of space-time Brownian motion on the curve f. Let be the class of all functions whose Hölder norm of order α is less than or equal to 1. We show that the supremum of L^f₁ over f in is finite if α>1/2 and infinite if α<1/2.     

On maximum increase and decrease of Brownian motion

The joint distribution of maximum increase and decrease for Brownian motion up to an independent exponential time is computed. This is achieved by decomposing the Brownian path at the hitting times of the infimum and the supremum before the exponential time. It is seen that an important element in our formula is the distribution of the maximum decrease for the three-dimensional Bessel process with drift started from 0 and stopped at the first hitting of a given level. From the joint distribution of the maximum increase and decrease it is possible to calculate the correlation coefficient between these at a fixed time and this is seen to be .     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

Particle filters with random resampling times

Particle filters are numerical methods for approximating the solution of the filtering problem which use systems of weighted particles that (typically) evolve according to the law of the signal process. These methods involve a corrective/resampling procedure which eliminates the particles that become redundant and multiplies the ones that contribute most to the resulting approximation. The correction is applied at instances in time called resampling/correction times. Practitioners normally use certain overall characteristics of the approximating system of particles (such as the effective sample size of the system) to determine when to correct the system. As a result, the resampling times are random. However, in the continuous time framework, all existing convergence results apply only to particle filters with deterministic correction times. In this paper, we analyse (continuous time) particle filters where resampling takes place at times that form a sequence of (predictable) stopping times. We prove that, under very general conditions imposed on the sequence of resampling times, the corresponding particle filters converge. The conditions are verified when the resampling times are chosen in accordance to the effective sample size of the system of particles, the coefficient of variation of the particles’ weights and, respectively, the (soft) maximum of the particles’ weights. We also deduce central-limit theorem type results for the approximating particle system with random resampling times.     

Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups

This paper studies drawdown and drawup processes in a general diffusion model. The main result is a formula for the joint distribution of the running minimum and the running maximum of the process stopped at the time of the first drop of size a$a$. As a consequence, we obtain the probabilities that a drawdown of size a$a$ precedes a drawup of size b$b$ and vice versa. The results are applied to several examples of diffusion processes, such as drifted Brownian motion, Ornstein–Uhlenbeck process, and Cox–Ingersoll–Ross process.     

Some inequalities for strong mixing random variables with applications to density estimation

In this paper, we establish an inequality of the characteristic functions for strongly mixing random vectors, by which, an upper bound is provided for the supremum of the absolute value of the difference of two multivariate probability density functions based on strongly mixing random vectors. As its application, we consider the consistency and asymptotic normality of a kernel estimate of a density function under strong mixing. Our results generalize some known results in the literature.     

The radial part of Brownian motion II. Its life and times on the cut locus

Summary This paper is a sequel to Kendall (1987), which explained how the Itô formula for the radial part of Brownian motionX on a Riemannian manifold can be extended to hold for all time including those times a whichX visits the cut locus. This extension consists of the subtraction of a correction term, a continuous predictable non-decreasing processL which changes only whenX visits the cut locus. In this paper we derive a representation onL in terms of measures of local time ofX on the cut locus. In analytic terms we compute an expression for the singular part of the Laplacian of the Riemannian distance function. The work uses a relationship of the Riemannian distance function to convexity, first described by Wu (1979) and applied to radial parts of -martingales in Kendall (1993).The first author's research was supported by a visiting fellowship awarded by the UK Science and Engineering Council, by travel funds provided by a European Community SCIENCE initiative, by the Max-Planck-Institute of Bonn, and by a grant from NSA     

Selector processes on classes of sets

Consider the random subset X of ℕ obtained by selecting independently each integer with a probability δ. Consider a finite class of finite sets. We describe a combinatorial quantity that is of the same order as We then give a related result allowing to compute the supremum of the empirical process on a class of sets. Work partially supported by an NSF grant.     

Asymptotic results for time-changed Lévy processes sampled at hitting times

We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to ε. For a wide class of Lévy processes, we introduce a renormalization depending on ε, under which the Lévy process converges in law to an α-stable process as ε goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal-Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.     

Riemann-integration and a new proof of the Bichteler–Dellacherie theorem

We give a new proof of the celebrated Bichteler–Dellacherie theorem, which states that a process S$S$ is a good integrator if and only if it is the sum of a local martingale and a finite-variation process. As a corollary, we obtain a characterization of semimartingales along the lines of classical Riemann integrability.