Asymptotic results for exponential functionals of Lévy processes

The asymptotic behavior of expectations of some exponential functionals of a Lévy process is studied. The key point is the observation that the asymptotics only depend on the sample paths with slowly decreasing local infimum. We give not only the convergence rate but also the expression of the limiting coefficient. The latter is given in terms of some transformations of the Lévy process based on its renewal function. As an application, we give an exact evaluation of the decay rate of the survival probability of a continuous-state branching process in random environment with stable branching mechanism.     

Exponential convergence in probability for empirical means of Lévy processes

Let （Xt）t≥0 be a Lévy process taking values in R^d with absolutely continuous marginal distributions. Given a real measurable function f on R^d in Kato＇s class, we show that the empirical mean 1/t ∫ f（Xs）ds converges to a constant z in probability with an exponential rate if and only if f has a uniform mean z. This result improves a classical result of Kahane et al. and generalizes a similar result of L. Wu from the Brownian Motion to general Lévy processes.     

Lévy processes and quasi-shuffle algebras

We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfil the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed quadratic covariation processes and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Lévy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Lévy processes forms a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Lévy processes.     

Exponential functional of Lévy processes: Generalized Weierstrass products and Wiener–Hopf factorization

In this note, we state a representation of the Mellin transform of the exponential functional of Lévy processes in terms of generalized Weierstrass products. As by-product, we obtain a multiplicative Wiener–Hopf factorization generalizing previous results obtained by Patie and Savov (2012) [14] as well as smoothness properties of its distribution.     

Abrupt Lévy processes

Among Lévy processes with unbounded variation, we distinguish the abrupt ones, which are characterised by infinitely sharp extrema. Stable processes with parameter α>1 and creeping Lévy processes are abrupt. We give a characterisation of abrupt processes and study their Dini derivatives at all points of their trajectories.     

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.     

Structural properties of reflected Lévy processes

This paper considers a number of structural properties of reflected Lévy processes, where both one-sided reflection (at 0) and two-sided reflection (at both 0 and K>0) are examined. With V _t being the position of the reflected process at time t, we focus on the analysis of $\zeta(t):=\mathbb{E}V_{t}$ and $\xi(t):=\mathbb{V}\mathrm{ar}V_{t}$ . We prove that for the one- and two-sided reflection, ζ(t) is increasing and concave, whereas for the one-sided reflection, ξ(t) is increasing. In most proofs we first establish the claim for the discrete-time counterpart (that is, a reflected random walk), and then use a limiting argument. A key step in our proofs for the two-sided reflection is a new representation of the position of the reflected process in terms of the driving Lévy process.     

Harmonic analysis of additive Lévy processes

Let X ₁, . . . ,X _N denote N independent d-dimensional Lévy processes, and consider the N-parameter random field $$\mathfrak{X}(t) := X_1(t_1)+\cdots+ X_N(t_N).$$ First we demonstrate that for all nonrandom Borel sets ${F\subseteq{{\bf R}^d}}$ , the Minkowski sum ${\mathfrak{X}({{\bf R}^{N}_{+}})\oplus F}$ , of the range ${\mathfrak{X}({{\bf R}^{N}_{+}})}$ of ${\mathfrak{X}}$ with F, can have positive d-dimensional Lebesgue measure if and only if a certain capacity of F is positive. This improves our earlier joint effort with Yuquan Zhong by removing a certain condition of symmetry in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003). Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical (non-probabilistic) harmonic analysis that might be of independent interest. As was shown in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003), the potential theory of the type studied here has a large number of consequences in the theory of Lévy processes. Presently, we highlight a few new consequences.     

The multifractal nature of Lévy processes

We show that the sample paths of most Lévy processes are multifractal functions and we determine their spectrum of singularities. Received: 21 February 1997 / Revised version: 27 July 1998     

Fluctuations of Omega-killed spectrally negative Lévy processes

In this paper we solve the exit problems for (reflected) spectrally negative Lévy processes, which are exponentially killed with a killing intensity dependent on the present state of the process and analyze respective resolvents. All identities are given in terms of new generalizations of scale functions. For the particular cases $\omega \left(x\right)=q$ and $\omega \left(x\right)=q{\mathbf{1}}_{(a,b)}\left(x\right)$, we obtain results for the classical exit problems and the Laplace transforms of the occupation times in a given interval, until first passage times, respectively. Our results can also be applied to find the bankruptcy probability in the so-called Omega model, where bankruptcy occurs at rate $\omega \left(x\right)$ when the Lévy surplus process is at level $x<0$. Finally, we apply these results to obtain some exit identities for spectrally positive self-similar Markov processes. The main method throughout all the proofs relies on the classical fluctuation identities for Lévy processes, the Markov property and some basic properties of a Poisson process.     

Optimal stopping of Markov switching Lévy processes

We consider a finite time horizon optimal stopping of a regime-switching Lévy process. We prove that the value function of the optimal stopping problem can be characterized as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman variational inequalities.     

Asymptotical properties of distributions of isotropic Lévy processes

In this paper, we establish the precise asymptotic behaviors of the tail probability and the transition density of a large class of isotropic Lévy processes when the scaling order is between 0 and 2 including 2. We also obtain the precise asymptotic behaviors of the tail probability of subordinators when the scaling order is between 0 and 1 including 1.The asymptotic expressions are given in terms of the radial part of characteristic exponent $\psi$ and its derivative. In particular, when $\psi \left(\lambda \right)?\frac{\lambda }{2}{\psi }^{\prime }\left(\lambda \right)$ varies regularly, as $\frac{t\psi {\left(r^{?1}\right)}^2}{\psi \left(r^{?1}\right)?{\left(2r\right)}^{?1}{\psi }^{\prime }\left(r^{?1}\right)}\to 0$ the tail probability $–(\left|X_t\right|\ge r)$ is asymptotically equal to a constant times $t(\psi \left(r^{?1}\right)?{\left(2r\right)}^{?1}{\psi }^{\prime }\left(r^{?1}\right)).$     

On hypoellipticity of generators of Lévy processes

We give a sufficient condition on a Lévy measure μ which ensures that the generator L of the corresponding pure jump Lévy process is (locally) hypoelliptic, i.e., \(\mathop {\mathrm {sing\,supp}}u\subseteq \mathop {\mathrm {sing\,supp}}Lu\) for all admissible u. In particular, we assume that \(\mu|_{\mathbb {R}^{d}\setminus \{0\}}\in C^{\infty}(\mathbb {R}^{d}\setminus \{0\})\). We also show that this condition is necessary provided that \(\mathop {\mathrm {supp}}\mu\) is compact.     

Measure preserving transformations of jump Lévy processes

Let x(t),t ? [ 0,1 ] \xi (t),t \in \left[ {0,1} \right] , be a jump Lévy process. By P_x {\mathcal{P}_\xi } we denote the law of in the Skorokhod space \mathbbD {\mathbb{D}} [0, 1]. Under some nondegeneracy condition on the Lévy measure Λ of the process, we construct a group of P_x {\mathcal{P}_\xi } -preserving transformations of the space \mathbbD {\mathbb{D}} [0, 1]. Bibliography: 10 titles.     

Right inverses of Lévy processes and stationary stopped local times

If X is a symmetric Lévy process on the line, then there exists a non-decreasing, càdlàg process H such that X(H(x)) = x for all x≥ 0 if and only if X is recurrent and has a non-trivial Gaussian component. The minimal such H is a subordinator K. The law of K is identified and shown to be the same as that of a linear time change of the inverse local time at 0 of X. When X is Brownian motion, K is just the usual ladder times process and this result extends the classical result of Lévy that the maximum process has the same law as the local time at 0. Write G _t for last point in the range of K prior to t. In a parallel with classical fluctuation theory, the process Z := (X _t −X _Gt ) _{t ≥0} is Markov with local time at 0 given by (X _Gt ) _{t ≥0}. The transition kernel and excursion measure of Z are identified. A similar programme is outlined for Lévy processes on the circle. This leads to the construction of a stopping time such that the stopped local times constitute a stationary process indexed by the circle. Received: 7 September 1999 / Revised version: 9 November 1999 / Published online: 8 August 2000     

Smooth properties for semigroups of Lévy processes and their application

Explicit smooth properties for the semigroup of Lévy processes are derived in terms of its symbol. As an application, we obtain new sufficient conditions for the strong Feller property of stochastic differential equations driven by the additive Lévy process.     

Maximum loss and maximum gain of spectrally negative Lévy processes

The joint distribution of the maximum loss and the maximum gain is obtained for a spectrally negative Lévy process until the passage time of a given level. Their marginal distributions up to an independent exponential time are also provided. The existing formulas for Brownian motion with drift are recovered using the particular scale functions.