Local times of stochastic processes with positive definite bivariate densities

A new class of stochastic processes, called processes of positive bivariate type, is defined. Such a process is typically one whose bivariate density functions are positive definite, at least for pairs of time points which are sufficiently mutually close. The class includes stationary Gaussian processes and stationary reversible Markov processes, and is closed under the operations of composition and convolution. The purpose of this work is to show that the local times of such processes can be investigated in a natural way. One of the main contributions is an orthogonal expansion of the local time which is new even in the well-studied stationary Gaussian case. The basic tool in its construction is the Lancaster-Sarmanov expansion of a bivariate density in a series of canonical correlations and canonical variables.     

Local time–space stochastic calculus for Lévy processes

We develop a stochastic calculus on the plane with respect to the local times of a large class of Lévy processes. We can then extend to these Lévy processes an Itô formula that was established previously for Brownian motion. Our method provides also a multidimensional version of the formula. We show that this formula generates many “Itô formulas” that fit various problems. In the special case of a linear Brownian motion, we recover a recently established Itô formula that involves local times on curves. This formula is already used in financial mathematics.     

Gaussian processes with biconvex covariances

Let R(s, t) be a continuous, nonnegative, real valued function on a ≤ s ≤ t ≤ b. Suppose $\text{?R}\text{?s}\text{≥ 0}$, $\text{?R}\text{?t}\text{≤ 0}$, and $\text{?}{}^2\text{R}\text{?t}\text{?t ≤ 0}$ in the interior of the domain. Then the extension of R to a symmetric function on [a, b] × [a, b] is a covariance function. Such a covariance is called biconvex. Let X(t) be a Gaussian process with mean 0 and biconvex covariance. X has a representation as a sum of simple moving averages of white noises on the line and plane. The germ field of X at every point t is generated by X(t) alone. X is locally nondeterministic. Under an additional assumption involving the partial derivatives of R near the diagonal, the local time of the sample function exists and is jointly continuous almost surely, so that the sample function is nowhere differentiable.     

Weak Dirichlet processes with a stochastic control perspective

The motivation of this paper is to prove verification theorems for stochastic optimal control of finite dimensional diffusion processes without control in the diffusion term, in the case where the value function is assumed to be continuous in time and once differentiable in the space variable (C^0,1$C^{0,1}$) instead of once differentiable in time and twice in space (C^1,2$C^{1,2}$), like in the classical results. For this purpose, the replacement tool of the Itô formula will be the Fukushima–Dirichlet decomposition for weak Dirichlet processes. Given a fixed filtration, a weak Dirichlet process is the sum of a local martingale M$M$ plus an adapted process A$A$ which is orthogonal, in the sense of covariation, to any continuous local martingale. The decomposition mentioned states that a C^0,1$C^{0,1}$ function of a weak Dirichlet process with finite quadratic variation is again a weak Dirichlet process. That result is established in this paper and it is applied to the strong solution of a Cauchy problem with final condition.     

Study of dependence for some stochastic processes: Symbolic Markov copulae

We study dependence between components of multivariate (nice Feller) Markov processes: what conditions need to be satisfied by a multivariate Markov process so that its components are Markovian with respect to the filtration of the entire process and such that they follow prescribed laws? To answer this question, we introduce a symbolic approach, which is rooted in the concept of pseudo-differential operator (PDO). We investigate connections between dependence, in the sense described above, and the PDOs. In particular, we study the problem of constructing a multivariate nice Feller process with given marginal laws in terms of symbols of the related PDOs. This approach leads to relatively simple conditions, which provide solutions to this problem.     

The asymptotic behaviour of local times and occupation integrals of theN-parameter Wiener process in R d

Summary LetL(x, T),xR ^d ,TR ₊ ^N , be the local time of theN-parameter Wiener processW taking values inR ^d . Even in the distribution valued casedd2N,L can be described in a series representation by means of multiple Wiener-Ito integrals. This setting proves to be a good starting point for the investigation of the asymptotic behaviour ofL(x, T) as |x|0 and/orT and of related occupation integrals asT. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws forL(x, T) resp.X _T (f), and of the second order, i.e. normalized convergence laws forL(x, T)–E(L(x, T)) resp.X _T (f)–E(X _T (f)).This research was made during a stay at the LMU in München supported by DAAD     

A test for the independence of two Gaussian processes

A bivariate Gaussian process with mean 0 and covariance

$\text{Σ(s, t, p)=}\text{Σ}{}_{11}\text{(s, t)}\text{ρΣ}{}_{12}\text{(s, t)}\text{ρΣ}{}_{21}\text{(s, t)}\text{Σ}{}_{22}\text{(s, t)}$

is observed in some region Ω of R′, where {Σ_ij(s,t)} are given functions and p an unknown parameter. A test of H₀: p = 0, locally equivalent to the likelihood ratio test, is given for the case when Ω consists of p points. An unbiased estimate of p is given. The case where Ω has positive (but finite) Lebesgue measure is treated by spreading the p points evenly over Ω and letting p → ∞. Two distinct cases arise, depending on whether Δ_2,p, the sum of squares of the canonical correlations associated with Σ(s, t, 1) on $\text{Ω}{}^2$, remains bounded. In the case of primary interest as p → ∞, Δ_2,p → ∞, in which case $\text{p}\text{?}$ converges to p and the power of the one-sided and two-sided tests of H₀ tends to 1. (For example, this case occurs when Σ_ij(s, t) ≡ Σ₁₁(s, t).)     

Particle picture interpretation of some Gaussian processes related to fractional Brownian motion

We construct fractional Brownian motion, sub-fractional Brownian motion and negative sub-fractional Brownian motion by means of limiting procedures applied to some particle systems. These processes are obtained for full ranges of Hurst parameter.     

Functional limit theorems for generalized quadratic variations of Gaussian processes

In this paper, we establish functional convergence theorems for second order quadratic variations of Gaussian processes which admit a singularity function. First, we prove a functional almost sure convergence theorem, and a functional central limit theorem, for the process of second order quadratic variations, and we illustrate these results with the example of the fractional Brownian sheet (FBS). Second, we do the same study for the process of localized second order quadratic variations, and we apply the results to the multifractional Brownian motion (MBM).     

Existence of the linear prediction for Banach space valued Gaussian processes

The correspondence between Gaussian stochastic processes with values in a Banach space E and cylindrical processes which are related to them is studied. It is shown that the linear prediction of an E-valued Gaussian process is an E-valued random variable as well as the spectral measure of an E-valued Gaussian stationary process is a Gaussian random measure.     

Lévy area for Gaussian processes: A double Wiener-Itô integral approach

Let {X₁(t)}_0≤t≤1 and {X₂(t)}_0≤t≤1 be two independent continuous centered Gaussian processes with covariance functions R₁ and R₂. We show that if the covariance functions are of finite p-variation and q-variation respectively and such that p⁻¹+q⁻¹>1, then the Lévy area can be defined as a double Wiener-Itô integral with respect to an isonormal Gaussian process induced by X₁ and X₂. Moreover, some properties of the characteristic function of that generalised Lévy area are studied.     

Dependence orderings for some functionals of multivariate point processes

We study dependence orderings for functionals of k-variate point processes Φ and Ψ. We view the first process as a collection of counting measures, whereas the second as the sequences of interpoint distances. Subsequently, we establish regularity properties of stationary sequences which generalize known results for iid case. The theoretical results are illustrated by many special cases including comparison of multivariate sums and products, comparison of multivariate shock models and queueing systems.     

On almost sure limit inferior for B-valued stochastic processes and applications

Summary A criterion on almost sure limit inferior for the increments of B-valued stochastic processes is presented. Applications to processes of independent increments and to Gaussian processes with stationary increments are given. In particular, an exact limit inferior bound is established for increments of infinite series of independent Ornstein-Uhlenbeck processes.Work supported by an NSERC Canada grant at Carleton UniversityWork supported by the Fok Yingtung Education Foundation of China     

On the regularity of the distribution of the maximum of one-parameter Gaussian processes

The main result in this paper states that if a one-parameter Gaussian process has C ^2k paths and satisfies a non-degeneracy condition, then the distribution of its maximum on a compact interval is of class C ^k . The methods leading to this theorem permit also to give bounds on the successive derivatives of the distribution of the maximum and to study their asymptotic behaviour as the level tends to infinity. Received: 14 May 1999 / Revised version: 18 October 1999 / Published online: 14 December 2000     

Strong local nondeterminism and exact modulus of continuity for spherical Gaussian fields

In this paper, we are concerned with sample path properties of isotropic spherical Gaussian fields on ${\mathbb{S}}^2$. In particular, we establish the property of strong local nondeterminism of an isotropic spherical Gaussian field based on the high-frequency behaviour of its angular power spectrum; we then exploit this result to establish an exact uniform modulus of continuity for its sample paths. We also discuss the range of values of the spectral index for which the sample functions exhibit fractal or smooth behaviour.