Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes

We study the distribution and various properties of exponential functionals of hypergeometric Lévy processes. We derive an explicit formula for the Mellin transform of the exponential functional and give both convergent and asymptotic series expansions of its probability density function. As applications we present a new proof of some of the results on the density of the supremum of a stable process, which were recently obtained in Hubalek and Kuznetsov (Electron. Commun. Probab. 16:84–95, 2011) and Kuznetsov (Ann. Probab. 39(3):1027–1060, 2011). We also derive several new results related to (i) the entrance law of a stable process conditioned to stay positive, (ii) the entrance law of the excursion measure of a stable process reflected at its past infimum, (iii) the distribution of the lifetime of a stable process conditioned to hit zero continuously and (iv) the entrance law and the last passage time of the radial part of a multidimensional symmetric stable process.     

Equivalence of Non-Gaussian Linear Processes

Two random processes x_t and y_t on an index set G are said to be equivalent iffor any positive integer n and any t_1,t_2,…,t_n∈G, (x_(t_1),x_(t_2),…,x_(t_n)) and (y_(t1),y_(t2),…, y_(t_n)) have the same joint probability distributions. Note that x_t and y_t may betwo random processes on a probability space or on two different probability spaces. The Equivalence Theorem Let x_t and y_t be non-Gaussian linear processes ona countable abelian group G:     

Asymptotic Behavior of Generalized Risk Processes

In this paper, the asymptotic behavior of generalized risk processes without any moment assumptions on the controlling process is described.     

The Entrance Laws of Self-Similar Markov Processes and Exponential Functionals of Lévy Processes

We consider the asymptotic behavior of semi-stable Markov processes valued in ]0,[ when the starting point tends to 0. The entrance distribution is expressed in terms of the exponential functional of the underlying Lévy process which appears in Lamperti's representation of a semi-stable Markov process.     

On the Estimation of Parameters of Poisson Processes

Let(N(t),t≥0)be a poisson process on probability spaces,i.e.(N(t),t≥0)is stochastic process with independent increments under P_θ and satisfies (a)N(0)=0 a.s.P_θ, (b)For all.     

On Variation of Single Birth Processes

Suppose {X（t）; t≥ 0} is a single birth process with birth rate qii＋l （i 〉 0） and death rate qij （i 〉 j ≥ 0）. It is proved in this paper that （i） if there exists aconstant c≥ 0 such that b（i）-a（i）＋ci is nondecreasing with respect to i and a（i） ＋ u（i） - ci ≥ 0 （i≥ 0）, then VarX（t）-EX（t）≥-X（0）e^-2ct,t≥0, or （ii） if there exists a constant u（i） - c≥ 0 such that b（i）-a（i）＋ci is non-increasing with respect to i and a（i）＋u（i）-ci≤0（i≥0）,then VarX（t） - EX（t） ≤ -X（0）e^-2c,t ≥ 0 Hereb（i） = qii＋1, a（0） = 0, a（i） = ∑j=^ijqii-j （i≥ 1）, u（0） = u（1） =0 and u（i） = 1/2∑j=^ij（j - 1）qii-j （i ≥ 2） . This result covers the results for birth-death processes obtained in [7].     

Martingale Representation of Functionals of Lévy Processes

Abstract

The main focus of the paper is a Clark–Ocone–Haussman formula for Lévy processes. First a difference operator is defined via the Fock space representation of L ²(P), then from this definition a Clark–Ocone–Haussman type formula is derived. We also derive some explicit chaos expansions for some common functionals. Later we prove that the difference operator defined via the Fock space representation and the difference operator defined by Picard [Picard, J. Formules de dualitésur l'espace de Poisson. Ann. Inst. Henri Poincaré 1996, 32 (4), 509–548] are equal. Finally, we give an example of how the Clark–Ocone–Haussman formula can be used to solve a hedging problem in a financial market modelled by a Lévy process.     

The Uniqueness Problem of Non-Gaussian Linear Processes

Consider a linear time series which is represented aswhere u_t is an independent and identically distributed random series with Eu_t=0,Eu_t~2=σ~2>0,w_t is a square-summable sequence     

Hölder Continuity of Random Processes

For a Young function φ and a Borel probability measure m on a compact metric space (T,d) the minorizing metric is defined by

Large Deviations of Local Times of Lévy Processes

For X(t) a real-valued symmetric Lévy process, its characteristic function is E(e ^iX(t))=exp(–t()). Assume that is regularly varying at infinity with index 1<2. Let L ^x _t denote the local time of X(t) and L* _t =sup _xR L ^x _t . Estimates are obtained for P(L ⁰ _t y) and P(L* _t y) as y and t fixed.     

On the Moments of the Modulus of Continuity of Itô Processes

The modulus of continuity of a stochastic process is a random element for any fixed mesh size. We provide upper bounds for the moments of the modulus of continuity of Itô processes with possibly unbounded coefficients, starting from the special case of Brownian motion. References to known results for the case of Brownian motion and Itô processes with uniformly bounded coefficients are included. As an application, we obtain the rate of strong convergence of Euler–Maruyama schemes for the approximation of stochastic delay differential equations satisfying a Lipschitz condition in supremum norm.     

Occupation Times of Refracted Lévy Processes

A refracted Lévy process is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More precisely, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation $$\begin{aligned} {\mathrm{d}}U_t=-\delta \mathbf 1 _{\{U_t>b\}}{\mathrm{d}}t +{\mathrm{d}}X_t,\quad t\ge 0 \end{aligned}$$ where \(X=(X_t, t\ge 0)\) is a Lévy process with law \(\mathbb{P }\) and \(b,\delta \in \mathbb{R }\) such that the resulting process \(U\) may visit the half line \((b,\infty )\) with positive probability. In this paper, we consider the case that \(X\) is spectrally negative and establish a number of identities for the following functionals $$\begin{aligned} \int \limits _0^\infty \mathbf 1 _{\{U_t where \(\kappa ^+_c=\inf \{t\ge 0: U_t> c\}\) and \(\kappa ^-_a=\inf \{t\ge 0: U_t< a\}\) for \(a . Our identities extend recent results of Landriault et al. (Stoch Process Appl 121:2629–2641, 2011) and bear relevance to Parisian-type financial instruments and insurance scenarios.     

The Weighted Transience and Recurrence of Markov Processes

Transience and recurrence are among the most important concepts in Markov processes. In this paper, we study the transience and recurrence for right processes with a given weight function, and characterize them by potentials, excessive functions, first hitting times and last exit times of the process. We also study the properties of recurrent states.     

Ruelle''''s Inequality for the Entropy of Diffusion Processes

In this paper we prove the Ruelle's inequality for the entropy and Lyapunovexponents of diffusion processes, that is, suppose h_μ,λ(i)(x),m_i (x) are respectivelythe entropy, Lyapunov exponents and its multiplicity for the random diffeomorphismsarising from SDE:     

Continuity Conditions for a Class of Gaussian Chaos Processes Related to Continuous Additive Functionals of Lévy Processes

Let be sequences of real numbers which are symmetric in k. Let be independent sequences of independent normal random variables with mean zero and variance one. For each fixed choice of we consider

α-Transience and α-Recurrence of Right Processes

In this article, we mainly discuss some potential theory in the framework of right Markov processes. We introduce the concept of α-excessive function, α-recurrence and α-transience for right processes with α ≤ 0, and give a thorough investigation.     

Stochastic Relations of Random Variables and Processes

This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov processes, it suffices that the generators of the processes preserve some, not necessarily reflexive or transitive, subrelation of the order relation. The main contributions of the paper are: a functional characterization of stochastic relations, necessary and sufficient conditions for the preservation of stochastic relations, and an algorithm for finding subrelations preserved by probability kernels. The theory is illustrated with applications to hidden Markov processes, population processes, and queueing systems.     

Occupation Times of General Lévy Processes

For an arbitrary Lévy process X which is not a compound Poisson process, we are interested in its occupation times. We use a quite novel and useful approach to derive formulas for the Laplace transform of the joint distribution of X and its occupation times. Our formulas are compact, and more importantly, the forms of the formulas clearly demonstrate the essential quantities for the calculation of occupation times of X. It is believed that our results are important not only for the study of stochastic processes, but also for financial applications.