arXiv subjects

Pierre Patie

Publications and source records attributed to Pierre Patie.

32 records · Page 2Linked to original sources

arXiv2017

Intertwining, Excursion Theory and Krein Theory of Strings for Non-self-adjoint Markov Semigroups

In this paper, we start by showing that the intertwining relationship between two minimal Markov semigroups acting on Hilbert spaces implies that any recurrent extensions, in the sense of Itô, of these semigroups satisfy the same intertwining identity. Under mild additional assumptions on the intertwining operator, we prove that the converse also holds. This connection, which relies on the representation of excursion quantities as developed by Fitzsimmons and Getoor, enables us to give an interesting probabilistic interpretation of intertwining relationships between Markov semigroups via excursion theory: two such recurrent extensions that intertwine share, under an appropriate normalization, the same local time at the boundary point. Moreover, in the case when one of the (non-self-adjoint) semigroup intertwines with the one of a quasi-diffusion, we obtain an extension of Krein's theory of strings byshowing that its densely defined spectral measure is absolutely continuous with respect to the measure appearing in the Stieltjes representation of the Laplace exponent of the inverse local time. Finally, we illustrate our results with the class of positive self-similar Markov semigroups and also the reflected generalized Laguerre semigroups. For the latter, we obtain their spectral decomposition and provide, under some conditions, a perturbed spectral gap estimate for its convergence to equilibrium.

math.PR↗

arXiv2016

Spectral decomposition of fractional operators and a reflected stable semigroup

In this paper, we provide the spectral decomposition in Hilbert space of the $\mathcal{C}_0$-semigroup $P$ and its adjoint $\hatP$ having as generator, respectively, the Caputo and the right-sided Riemann-Liouville fractional derivatives of index $1<α<2$. These linear operators, which are non-local and non-self-adjoint, appear in many recent studies in applied mathematics and also arise as the infinitesimal generators of some substantial processes such as the reflected spectrally negative $α$-stable process. Our approach relies on intertwining relations that we establish between these semigroups and the semigroup of a Bessel type process whose generator is a self-adjoint second order differential operator. In particular, from this commutation relation, we characterize the positive real axis as the continuous point spectrum of $P$ and provide a power series representation of the corresponding eigenfunctions. We also identify the positive real axis as the residual spectrum of the adjoint operator $\hatP$ and elucidates its role in the spectral decomposition of these operators. By resorting to the concept of continuous frames, we proceed by investigating the domain of the spectral operators and derive two representations for the heat kernels of these semigroups. As a by-product, we also obtain regularity properties for these latter and also for the solution of the associated Cauchy problem.

math.PR↗

arXiv2016

Cauchy Problem of the non-self-adjoint Gauss-Laguerre semigroups and uniform bounds of generalized Laguerre polynomials

We propose a new approach to construct the eigenvalue expansion in a weighted Hilbert space of the solution to the Cauchy problem associated to Gauss-Laguerre invariant Markov semigroups that we introduce. Their generators turn out to be natural non-self-adjoint and non-local generalizations of the Laguerre differential operator. Our methods rely on intertwining relations that we establish between these semigroups and the classical Laguerre semigroup and combine with techniques based on non-harmonic analysis. As a by-product we also provide regularity properties for the semigroups as well as for their heat kernels. The biorthogonal sequences that appear in their eigenvalue expansion can be expressed in terms of sequences of polynomials, and they generalize the Laguerre polynomials. By means of a delicate saddle point method, we derive uniform asymptotic bounds that allow us to get an upper bound for their norms in weighted Hilbert spaces. We believe that this work opens a way to construct spectral expansions for more general non-self-adjoint Markov semigroups.

math.PR↗

arXiv2016

Spectral expansions of non-self-adjoint generalized Laguerre semigroups

We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these norms.

math.PR↗

arXiv2014

Boundary crossing identities for Brownian motion and some nonlinear ode's

We start by introducing a nonlinear involution operator which maps the space of solutions of Sturm-Liouville equations into the space of solutions of the associated equations which turn out to be nonlinear ordinary differential equations. We study some algebraic and analytical properties of this involution operator as well as some properties of a two-parameter family of operators describing the set of solutions of Sturm-Liouville equations. Next, we show how a specific composition of these mappings allows to connect, by means of a simple analytical expression, the law of the first passage time of a Brownian motion over a curve to a two-parameter family of curves. We offer three different proofs of this fact which may be of independent interests. In particular, one is based on the construction of parametric time-space harmonic transforms of the law of some Gauss-Markov processes. Another one, which is of algebraic nature, relies on the Lie group symmetry methods applied to the heat equation and reveals that our two-parameter transformation is the unique non-trivial one.

math.PR↗

arXiv2012

Optimal stopping problems for some Markov processes

In this paper, we solve explicitly the optimal stopping problem with random discounting and an additive functional as cost of observations for a regular linear diffusion. We also extend the results to the class of one-sided regular Feller processes. This generalizes the result of Beibel and Lerche [Statist. Sinica 7 (1997) 93-108] and [Teor. Veroyatn. Primen. 45 (2000) 657-669] and Irles and Paulsen [Sequential Anal. 23 (2004) 297-316]. Our approach relies on a combination of techniques borrowed from potential theory and stochastic calculus. We illustrate our results by detailing some new examples ranging from linear diffusions to Markov processes of the spectrally negative type.

math.PR↗

arXiv2012

A Wiener-Hopf Type Factorization for the Exponential Functional of Levy Processes

For a Lévy process $ξ=(ξ_t)_{t\geq0}$ drifting to $-\infty$, we define the so-called exponential functional as follows \[{\rm{I}}_ξ=\int_0^{\infty}e^{ξ_t} dt.\] Under mild conditions on $ξ$, we show that the following factorization of exponential functionals \[{\rm{I}}_ξ\stackrel{d}={\rm{I}}_{H^-} \times {\rm{I}}_{Y}\] holds, where, $\times $ stands for the product of independent random variables, $H^-$ is the descending ladder height process of $ξ$ and $Y$ is a spectrally positive Lévy process with a negative mean constructed from its ascending ladder height process. As a by-product, we generate an integral or power series representation for the law of ${\rm{I}}_ξ$ for a large class of Lévy processes with two-sided jumps and also derive some new distributional properties. The proof of our main result relies on a fine Markovian study of a class of generalized Ornstein-Uhlenbeck processes which is of independent interest on its own. We use and refine an alternative approach of studying the stationary measure of a Markov process which avoids some technicalities and difficulties that appear in the classical method of employing the generator of the dual Markov process.

math.PR↗

arXiv2011

Extended Factorizations of Exponential Functionals of Lévy Processes

In [16], under mild conditions, a Wiener-Hopf type factorization is derived for the exponential functional of proper Lévy processes. In this paper, we extend this factorization by relaxing a finite moment assumption as well as by considering the exponential functional for killed Lévy processes. As a by-product, we derive some interesting new distributional properties enjoyed by a large class of this random variable, such as the absolute continuity of its distribution and the smoothness, boundedness or complete monotonicity of its density. This type of results is then used to derive similar properties for the law of maxima and first passage time of some stable Lévy processes. Thus, for example, we show that for a large class of stable processes the first passage time has a bounded and non-increasing density on the positive half-line. We also generate many instances of integral or power series representations for the law of the exponential functional of Lévy processes with one or two-sided jumps. The proof of our main results requires different devices from the one developed in [16]. It relies in particular on a generalization of a transform recently introduced in [8] together with some extensions to killed Lévy process of Wiener-Hopf techniques.

math.PR↗

arXiv2010

A transformation for Lévy processes with one-sided jumps and applications

The aim of this work is to extend and study a family of transformations between Laplace exponents of Lévy processes which have been introduced recently in a variety of different contexts by Patie, Kyprianou and Patie, and, Gnedin, as well as in older work of Urbanik . We show how some specific instances of this mapping prove to be useful for a variety of applications.

math.PR↗

arXiv2010

Intertwining certain fractional derivatives

We obtain an intertwining relation between some Riemann-Liouville operators of order a in (1,2) connecting through a certain multiplicative identity in law the one-dimensional marginals of reflected completely asymmetric a-stable Lévy processes. An alternative approach based on recurrent extensions of positive self-similar Markov processes and exponential functionals of Lévy processes is also discussed.

math.PR↗

arXiv2010

Absolute ruin in the Ornstein-Uhlenbeck type risk model

We start by showing that the finite-time absolute ruin probability in the classical risk model with constant interest force can be expressed in terms of the transition probability of a positive Ornstein-Uhlenbeck type process, say X. Our methodology applies to the case when the dynamics of the aggregate claims process is a subordinator. From this expression, we easily deduce necessary and sufficient conditions for the infinite-time absolute ruin to occur. We proceed by showing that, under some technical conditions, the transition density of X admits a spectral type representation involving merely the limiting distribution of the process. As a by-product, we obtain a series expansions for the finite-time absolute ruin probability. On the way, we also derive, for the aforementioned risk process, the Laplace transform of the first-exit time from an interval from above. Finally, we illustrate our results by detailing some examples.

q-fin.CP↗

arXiv2009

Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes

We provide the increasing eigenfunctions associated to spectrally negative self-similar Feller semigroups, which have been introduced by Lamperti. These eigenfunctions are expressed in terms of a new family of power series which includes, for instance, the modified Bessel functions of the first kind and some generalizations of the Mittag-Leffler function. Then, we show that some specific combinations of these functions are Laplace transforms of self-decomposable or infinitely divisible distributions concentrated on the positive line with respect to the main argument, and, more surprisingly, with respect to a parameter of the process. In particular, this generalizes a result of Hartman (1976) obtained for the increasing solution of the Bessel differential equation. Finally, we compute, for some cases, the associated decreasing eigenfunctions and derive the Laplace transform of the exponential functionals of some spectrally negative Lévy processes with a negative first moment.

math.PR↗

arXiv2009

Law of the exponential functional of one-sided Lévy processes and Asian options

The purpose of this note is to describe, in terms of a power series, the distribution function of the exponential functional, taken at some independent exponential time, of a spectrally negative Lévy process ξwith unbounded variation. We also derive a Geman-Yor type formula for Asian options prices in a financial market driven by e^ξ.

math.PR↗

arXiv2009

Boundary crossing identities for diffusions having the time inversion property

We review and study a one-parameter family of functional transformations, denoted by $(S^{(β)})_{β\in \R}$, which, in the case $β<0$, provides a path realization of bridges associated to the family of diffusion processes enjoying the time inversion property. This family includes the Brownian motion, Bessel processes with a positive dimension and their conservative $h$-transforms. By means of these transformations, we derive an explicit and simple expression which relates the law of the boundary crossing times for these diffusions over a given function $f$ to those over the image of $f$ by the mapping $S^{(β)}$, for some fixed $β\in \mathbb{R}$. We give some new examples of boundary crossing problems for the Brownian motion and the family of Bessel processes. We also provide, in the Brownian case, an interpretation of the results obtained by the standard method of images and establish connections between the exact asymptotics for large time of the densities corresponding to various curves of each family.

math.PR↗