Various recent results on quantum Lévy processes are presented. The first part provides an introduction to the theory of Lévy processes on involutive bialgebras. The notion of independence used for these processes is tensor independence, which generalizes the notion of independence used in classical probability and corresponds to independent observables in quantum physics.
In quantum probability there exist other notions of independence and Lévy processes can also be defined for the five so-called universal independences. This is the topic of the second part. In particular, it is shown that boolean, monotone, and anti-monotone independence can be reduced to tensor independence.
Finally, in the third part, several classes of quantum Lévy processes of special interest are considered, e.g., Lévy processes on real Lie algebras or Brownian motions on braided spaces. Several applications of these processes are also presented.
We describe basic motivations behind quantum or noncommutative probability, introduce quantum Lévy processes on compact quantum groups, and discuss several aspects of the study of the latter in the example of quantum permutation groups. The first half of this paper is a survey on quantum probability, compact quantum groups, and Lévy processes on compact quantum groups. In the second half the theory is applied to quantum permutations groups. Explicit examples are constructed and certain classes of such Lévy processes are classified.
We present the Levy's distributional property for symmetric Levy processes with generating triplet $(0, 0,ν)$ or $(σ>0, γ, ν)$ where $ν$ is a symmetric measure on $R\backslash\{0\}$. This generalizes the classical Levy's theorem about Brownian motions with drift.
Semi-Levy process is an additive process with periodically stationary increments. In particular, it is a generalization of Levy process. The dichotomy of recurrence and transience of Levy processes is well known, but this is not necessarily true for general additive processes. In this paper, we prove the recurrence and transience dichotomy of semi-Levy processes. For the proof, we introduce a concept of semi-random walk and discuss its recurrence and transience properties. An example of semi-Levy process constructed from two independent Levy processes is investigated. Finally, we prove the law…
We show on- and off-diagonal upper estimates for the transition densities of symmetric Levy and Levy-type processes. To get the an-diagonal estimates we prove a Nash type inequality for the related Dirichlet form. For the off-diagonal estimates we assume that the characteristic function of a Levy (type) process is analytic, which allows to apply the complex analysis technique.
Which Levy processes satisfy Hunt's hypothesis (H) is a long-standing open problem in probabilistic potential theory. The study of this problem for one-dimensional Levy processes suggests us to consider (H) from the point of view of the sum of Levy processes. In this paper, we present theorems and examples on the validity of (H) for the sum of two independent Levy processes. We also give a novel condition on the Levy measure which implies (H) for a large class of one-dimensional Levy processes.
Upper estimates of densities of convolution semigroups of probability measures are given under explicit assumptions on the corresponding Lévy measure and the Lévy--Khinchin exponent.
In this paper, we study the existence of the transition densities of one-dimensional Lévy processes. Compared with past results, our results contain the Lévy processes whose Lévy symbols have logarithm behavior at infinity. Our results contain the Lévy symbol induced by the following Laplace exponent $ψ(ξ) := (\ln(1 + \ln(1 + \ln(\cdots \ln(1 + |ξ|)))))^ε$ ($n$ times), $0 < ε< 1$, $2 \le n$. We also show that $ψ(ξ)$ is a Lévy symbol with transition density.
In this article, we define the new concept of local coupling property for Markov processes and study its relationship with distributional properties of the transition probability. In the special case of Lévy processes we show that this property is equivalent to the absolute continuity of the transition probability and also provide a sufficient condition for it in terms of the Lévy measure. Our result is stronger than existing results for absolute continuity of Lévy distributions.
The inversion of a Levy measure was first introduced (under a different name) in Sato 2007. We generalize the definition and give some properties. We then use inversions to derive a relationship between weak convergence of a Levy process to an infinite variance stable distribution when time approaches zero and weak convergence of a different Levy process as time approaches infinity. This allows us to get self contained conditions for a Levy process to converge to an infinite variance stable distribution as time approaches zero. We formulate our results both for general Levy processes and for t…
We review the probabilistic properties of Ornstein-Uhlenbeck processes in Hilbert spaces driven by Lévy processes. The emphasis is on the different contexts in which these processes arise, such as stochastic partial differential equations, continuous-state branching processes, generalised Mehler semigroups and operator self-decomposable distributions. We also examine generalisations to the case where the driving noise is cylindrical.
We prove two martingale identities which involve exit times of Levy-driven Ornstein--Uhlenbeck processes. Using these identities we find an explicit formula for the Laplace transform of the exit time under the assumption that positive jumps of the Levy process are exponentially distributed.
Combinatorial Levy processes evolve on general state spaces of countable combinatorial structures. In this setting, the usual Levy process properties of stationary, independent increments are defined in an unconventional way in terms of the symmetric difference operation on sets. In discrete time, the description of combinatorial Levy processes gives rise to the notion of combinatorial random walks. These processes behave differently than random walks and Levy processes on other state spaces. Standard examples include processes on sets, graphs, and n-ary relations, but the framework permits fa…
Kuznetsov et al. (2011) and Kuznetsov and Pardo (2013) introduced the family of Hypergeometric Lévy processes. They appear naturally in the study of fluctuations of stable processes when one analyses stable processes through the theory of positive self-similar Markov processes. Hypergeometric Lévy processes are defined through their characteristic exponent, which, as a complex-valued function, has four independent parameters. Kyprianou et al. (2014) showed that the definition of a Hypergeometric Lévy process could be taken to include a greater range of the aforesaid parameters than originally …
We completely describe the size and large intersection properties of the Holder singularity sets of Levy processes. We also study the set of times at which a given function cannot be a modulus of continuity of a Levy process. The Holder singularity sets of the sample paths of certain random wavelet series are investigated as well.
In this paper we prove the uniform boundary Harnack principle in general open sets for harmonic functions with respect to a large class of rotationally symmetric purely discontinuous Lévy processes.
We find necessary and sufficient conditions for almost sure finiteness of integral functionals of spectrally positive Lévy processes. Via Lamperti type transforms, these results can be applied to obtain new integral tests on extinction and explosion behaviors for a class of continuous-state nonlinear branching processes.
In this paper we prove a criterion of convergence in distribution in Skorokhod space. We apply this criterion to some special Levy processes and obtain almost-sure versions of limit theorems for these processes.
We ask for necessary and sufficient conditions for almost sure finiteness of the perpetual integrals of a Levy process. Zero-one laws are already known for Brownian motion with drift and spectrally one-sided Levy processes. Under the assumption that local times exist, we use fluctuation theory and Jeulin's lemma to prove the zero-one law.
Motivated by the recent results of Nualart and Xu \cite{Nualart} concerning limits laws for occupation times of one dimensional symmetric stable processes, this paper proves a decomposition for functionals of one dimensional symmetric Lévy processes under certain conditions on the characteristic exponent and computes the moments of the decomposition.
Exponential functionals of Lévy processes appear as stationary distributions of generalized Ornstein-Uhlenbeck (GOU) processes. In this paper we obtain the infinitesimal generator of the GOU process and show that it is a Feller process. Further we use these results to investigate properties of the mapping Φ, which maps two independent Lévy processes to their corresponding exponential functional, where one of the processes is assumed to be fixed. We show that in many cases this mapping is injective and give the inverse mapping in terms of (Lévy) characteristics. Also, continuity of Φis treated …
In the paper, we consider a type of stochastic differential equations driven by G-Lévy processes. We prove that a kind of their additive functionals has path independence and extend some known results.
In this article we consider the Levy processes and the corresponding semigroup. We represent the generator of this semigroup in a convolution form. Using the obtained convolution form and the theory of integral equations we investigate the properties of a wide class of Levy processes (potential, quasi-potential, the probability of the Levy process remaining within the given domain, long time behavior, stable processes). We analyze in detail a number of concrete examples of the Levy processes (stable processes, the variance damped Levy processes, the variance gamma processes, the normal Gaussia…
Minimal thinness is a notion that describes the smallness of a set at a boundary point. In this paper, we provide tests for minimal thinness at finite and infinite minimal Martin boundary points for a large class of purely discontinuous symmetric Lévy processes.
Cylindrical probability measures are finitely additive measures on Banach spaces that have sigma-additive projections to Euclidean spaces of all dimensions. They are naturally associated to notions of weak (cylindrical) random variable and hence weak (cylindrical) stochastic processes. In this paper we focus on cylindrical Levy processes. These have (weak) Levy-Ito decompositions and an associated Levy-Khintchine formula. If the process is weakly square integrable, its covariance operator can be used to construct a reproducing kernel Hilbert space in which the process has a decomposition as an…
Under some mild assumptions on the Levy measure and the symbol we obtain gradient estimates of Dirichlet heat kernels for pure-jump isotropic unimodal Levy processes in $R^d$.
In this paper, an integration by parts formula was derived for jump processes on Hilbert spaces. Using this formula, we investigated derivative formula and exponential ergodicity for nonlinear SPDEs driven by purely jump processes.
These lectures notes aim at introducing Lévy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of Lévy processes. We analyze a `toy' example of a Lévy process, viz. a Lévy jump-diffusion, which yet offers significant insight into the distributional and path structure of a Lévy process. Then, we present several important results about Lévy processes, such as infinite divisibility and the Lévy-Khintchine formula, the Lévy-Itô decomposition, the Itô formula for Lévy processes and Girsanov's transformation. Some (sketc…
We consider natural exponential families of Levy processes with randomized parameter. Such processes are Markov, and under suitable assumptions, pairs of such processes with shared randomization can be stitched together into a single harness. The stitching consists of deterministic reparametrization of the time for both processes, so that they run on adjacent time intervals, and of the choice of the appropriate law at the boundary. Processes in the Levy-Meixner class have an additional property that they are quadratic harnesses, and in this case stitching constructions produce quadratic harnes…
The last couple of years has seen a remarkable number of new, explicit examples of the Wiener-Hopf factorization for Levy processes where previously there had been very few. We mention in particular the many cases of spectrally negative Levy processes, hyper-exponential and generalized hyper-exponential Levy processes, Lamperti-stable processes, Hypergeometric processes, Beta-processes and Theta-processes. In this paper we introduce a new family of Levy processes, which we call Meromorphic Levy processes, or just M-processes for short, which overlaps with many of the aforementioned classes. A …
We determine the rate of decrease of the right tail distribution of the exponential functional of a Levy process with a convolution equivalent Levy measure. Our main result establishes that it decreases as the right tail of the image under the exponential function of the Levy measure of the underlying Levy process. The method of proof relies on fluctuation theory of Levy processes and an explicit path-wise representation of the exponential functional as the exponential functional of a bivariate subordinator. Our techniques allow us to establish rather general estimates of the measure of the ex…
The paper studies the rate of convergence of the weak Euler approximation for solutions to SDEs driven by Levy processes, with Hoelder-continuous coefficients. It investigates the dependence of the rate on the regularity of coefficients and driving processes. The equation considered has a non-degenerate main part driven by a spherically-symmetric stable process.
Suppose Xt is either a regular exponential type Levy process or a Levy process with a bounded variation jumps measure. The distribution of the extrema of Xt play a crucial role in many financial and actuarial problems. This article employs the well known and powerful Riemann-Hilbert technique to derive the characteristic functions of the extrema for such Levy processes. An approximation technique along with several examples is given.
In this paper, we consider a large class of purely discontinuous rotationally symmetric Levy processes. We establish sharp two-sided estimates for the transition densities of such processes killed upon leaving an open set D. When D is a κ-fat open set, the sharp two-sided estimates are given in terms of surviving probabilities and the global transition density of the Levy process. When D is a C^{1, 1} open set and the Levy exponent of the process is given by Ψ(ξ)= ϕ(|ξ|^2) with ϕbeing a complete Bernstein function satisfying a mild growth condition at infinity, our two-sided estimates are expl…
The characteristic measure of excursions away from a regular point is studied for a class of symmetric Lévy processes without Gaussian part. It is proved that the harmonic transform of the killed process enjoys Feller property. The result is applied to prove extremeness of the excursion measure and to prove several sample path behaviors of the excursion and the $ h $-path processes.
We establish a connection between the scattering inverse problem and the determination of the distribution of the position of the Levy process at the exit time of a bounded interval in term of its Levy exponent.
We prove the intrinsic ultracontractivity for the semigroup generated by a large class of symmetric Levy processes such that the Levy measure satisfies some conditions in the neighborhood of 0, killed on exiting a bounded and connected Lipschitz domain.
We construct superharmonic functions and give sharp bounds for the expected exit time and probability of survival for isotropic unimodal Lévy processes
For a broad class of the Levy processes the new form (convolution type) of the infinitesimal generators is introduced. It leads to the new notions: a truncated generator, a quasi-potential. The probability of the Levy process remaining within the given domain is estimated.
The pricing of options in exponential Levy models amounts to the computation of expectations of functionals of Levy processes. In many situations, Monte-Carlo methods are used. However, the simulation of a Levy process with infinite Levy measure generally requires either to truncate small jumps or to replace them by a Brownian motion with the same variance. We will derive bounds for the errors generated by these two types of approximation.
We give upper and lower estimates of densities of convolution semigroups of probability measures under explicit assumptions on the corresponding Levy measure and the Levy--Khinchin exponent. We obtain also estimates of derivatives of densities.
We show that the general Lévy process can be embedded in a suitable Fock space, classified by cocycles of the real line regarded as a group, ${\bf R}$. The formula of de Finetti corresponds to coboundaries. Kolmogorov's processes correspond to cocycles of which the derivatives are cocycles of the Lie algebra of ${\bf R}$. Lévy's formula gives the most general cocycle possible.
We extend the concept of packing dimension profiles, due to Falconer and Howroyd (1997) and Howroyd (2001), and use our extension in order to determine the packing dimension of an arbitrary image of a general Levy process.
In this note we apply the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Levy processes from Kuznetsov et al. [17] to path functionals, in particular first passage times, overshoots, undershoots and the last maximum before the passage time. Such functionals have many applications, for instance in finance (the pricing of exotic options in a Levy model) and insurance (ruin time, debt at ruin and related quantities for a Levy insurance risk process). The technique works for any Levy process whose running infimum and supremum evaluated at an independent exponential ti…
Every quantum Levy process with a bounded stochastic generator is shown to arise as a strong limit of a family of suitably scaled quantum random walks.
We extend the idea of tempering stable Levy processes to tempering more general classes of Levy processes. We show that the original process can be decomposed into the sum of the tempered process and an independent point process of large jumps. We then use this to set up a rejection sampling algorithm for sampling from the tempered process. A small scale simulation study is given to help understand the performance of this algorithm.
The classical notion of Lévy process is generalized to one that takes as its values probabilities on a first order model equipped with a commutative semigroup. This is achieved by applying a convolution product on definable probabilities and the infinite divisibility with respect to it.
A new approach to solve the continuous-time stochastic inventory problem using the fluctuation theory of Levy processes is developed. This approach involves the recent developments of the scale function that is capable of expressing many fluctuation identities of spectrally one-sided Levy processes. For the case with a fixed cost and a general spectrally positive Levy demand process, we show the optimality of an (s,S)-policy. The optimal policy and the value function are concisely expressed via the scale function. Numerical examples under a Levy process in the beta-family with jumps of infinit…
We give conditions under which the tail probability of the supremum over unit interval of a Levy process with light tail is equivalent to the tail of the value of the process at the right endpoint.
In our previous paper (ArXiv:1306.1492) we have proved that a representation of the infinitesimal generators $L$ for Levy processes $X_t$ can be written down in a convolution type form. For the case of non-summable Levy measures we constructed the quasi-potential operators $B$ and investigated the long time behavior of $X_t$. In the present paper we consider Levy processes $X_t$ with summable Levy measures. In this case the form of the quasi-potential operators $B$ essentially differs from the form in the case of non-summable Levy measures. We use this new form in order to study the long time …
Using the method of Krylov's estimates, we prove the existence of weak solutions of stochastic differential equations driven by purely discontinuous Levy processes satisfying an additional assumption. The diffusion coefficient is assumed to be one and the time-dependent drift is measurable and bounded.
We prove a version of the Feynman-Kac formula for Levy processes and integro-differential operators, with application to the momentum representation of suitable quantum (Euclidean) systems whose Hamiltonians involve Lévy-type potentials. Large deviation techniques are used to obtain the limiting behavior of the systems as the Planck constant approaches zero. It turns out that the limiting behavior coincides with fresh aspects of the semiclassical limit of (Euclidean) quantum mechanics. Non-trivial examples of Levy processes are considered as illustrations and precise asymptotics are given for …
