In this paper, we study the compressibility of random processes and fields, called generalized Lévy processes, that are solutions of stochastic differential equations driven by $d$-dimensional periodic Lévy white noises. Our results are based on the estimation of the Besov regularity of Lévy white noises and generalized Lévy processes. We show in particular that non-Gaussian generalized Lévy processes are more compressible in a wavelet basis than the corresponding Gaussian processes, in the sense that their $n$-term approximation error decays faster. We quantify this compressibility in terms of the Blumenthal-Getoor index of the underlying Lévy white noise.
We derive a necessary and sufficient condition for stochastic processes to have almost periodic finite dimensional distributions; in particular, we obtain characterizations for infinitely divisible processes to be almost periodic in terms of their characteristic triplets. Furthermore, we derive conditions when the process $(X_t)_{t\in\R}$ defined by the stochastic integral $X_t:= \int_{\R^d} f(t,s) dL(s)$ is almost periodic stationary and also when it is almost periodic in probability, where $f(t,\cdot)\in L^1(\R^d,\R)\cap L^2(\R^d,\R)$ is deterministic and $L$ is a Lévy basis. Moreover, we di…
Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincaré sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only has equally placed simple eigenvalues including $0$ on the imaginary axis, then the periodic measure is PS-ergodic and has positive minimum period. Conversely if the periodic measure with the positive minimum period is PS-mixing, then the infinitesimal generator only has equally placed simp…
We consider a family of random locations, called intrinsic location functionals, of periodic stationary processes. This family includes but is not limited to the location of the path supremum and first/last hitting times. We first show that the set of all possible distributions of intrinsic location functionals for periodic stationary processes is the convex hull generated by a specific group of distributions. We then focus on two special subclasses of these random locations. For the first subclass, the density has a uniform lower bound; for the second subclass, the possible distributions are …
Given a spectrally negative Lévy process and independent Poisson observation times, we consider a periodic barrier strategy that pushes the process down to a certain level whenever it is above it. We also consider the versions with additional classical reflection above and/or below. Using scale functions and excursion theory, various fluctuation identities are computed in terms of the scale function. Applications in de Finetti's dividend problems are also discussed.
This paper deals with ergodic theorems for particular time-inhomogeneous Markov processes, whose the time-inhomogeneity is asymptotically periodic. Under a Lyapunov/minorization condition, it is shown that, for any measurable bounded function $f$, the time average $\frac{1}{t} \int_0^t f(X_s)ds$ converges in $\mathbb{L}^2$ towards a limiting distribution, starting from any initial distribution for the process $(X_t)_{t \geq 0}$. This convergence can be improved to an almost sure convergence under an additional assumption on the initial measure. This result will be then applied to show the exis…
We study the hydrodynamic limits of the simple exclusion processes and the zero range processes on crystal lattices. For a periodic realization of crystal lattice, we derive the hydrodynamic limit for the exclusion processes and the zero range processes, which depends on both the structure of crystal lattice and the periodic realization. Even through the crystal lattices have inhomogeneous local structure, for all periodic realizations, we apply the entropy method to derive the hydrodynamic limits. Also, we discuss how the limit equation depends on the choices of the realizations.
In this paper, we investigate periodic solutions of regime-switching jump diffusions. We first show the well-posedness of solutions to the SDEs corresponding to the hybrid system. Then, we derive the strong Feller property and irreducibility of the associated time-inhomogeneous semigroups. Finally, we establish the existence and uniqueness of periodic solutions. Concrete examples are presented to illustrate the results.
This work contributes to the theory and applications of Hawkes processes. We introduce and examine a new class of Hawkes processes that we call generalized Hawkes processes, and their special subclass -- the generalized multivariate Hawkes processes (GMHPs). GMHPs are multivariate marked point processes that add an important feature to the family of the (classical) multivariate Hawkes processes: they allow for explicit modelling of simultaneous occurrence of excitation events coming from different sources, i.e. caused by different coordinates of the multivariate process. We study the issue of …
The paper deals with a family of jump Markov process defined in a medium with a periodic or locally periodic microstructure. We assume that the generator of the process is a zero order convolution type operator with rapidly oscillating locally periodic coefficient and, under natural ellipticity and localization conditions, show that the family satisfies the large deviation principle in the path space equipped with Skorokhod topology. The corresponding rate function is defined in terms of a family of auxiliary periodic spectral problems.
We propose periodic Macdonald processes as a $(q,t)$-deformation of periodic Schur processes and a periodic analogue of Macdonald processes. It is known that, in the theory of stochastic processes related to a family of symmetric functions, the Cauchy-like identity gives an explicit expression of a partition function. We compute the partition functions of periodic Macdonald processes relying on the free field realization of the Macdonald theory. We also study several families of observables for periodic Macdonald processes and give formulas of their moments. The technical tool is the free fiel…
Maximising dividends is one classical stability criterion in actuarial risk theory. Motivated by the fact that dividends are paid periodically in real life, $\textit{periodic}$ dividend strategies were recently introduced (Albrecher, Gerber and Shiu, 2011). In this paper, we incorporate fixed transaction costs into the model and study the optimal periodic dividend strategy with fixed transaction costs for spectrally negative Lévy processes. The value function of a periodic $(b_u,b_l)$ strategy is calculated by means of exiting identities and Itô's excusion when the surplus process is of unboun…
The paper presents linear integral predictors for continuous time high-frequency signals with a a finite spectrum gap. The predictors are based on approximation of a complex valued periodic exponential (complex sinusoid) by rational polynomials.
The theory of sparse stochastic processes offers a broad class of statistical models to study signals. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by white Lévy noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized Lévy process-from Gaussian to sparse-can be understood as the limit in law of a sequence of generalized Poisson processes. This …
We first study the drift parameter estimation of the fractional Ornstein-Uhlenbeck process (fOU) with periodic mean for every $\frac{1}{2}<H<1$. More precisely, we extend the consistency proved in \cite{DFW} for $\frac{1}{2}<H<\frac{3}{4}$ to the strong consistency for any $\frac{1}{2}<H<1$ on the one hand, and on the other, we also discuss the asymptotic normality given in \cite{DFW}. In the second main part of the paper, we study the strong consistency and the asymptotic normality of the fOU of the second kind with periodic mean for any $\frac{1}{2}<H<1$.
