Pierre Patie

Representations of the first hitting time density of an Ornstein-Uhlenbeck process

ABSTRACT Three expressions are provided for the first hitting time density of an Ornstein-Uhlenbeck process to reach a fixed level. The first hinges on an eigenvalue expansion involving zeros of the parabolic cylinder functions. The second is an integral representation involving some special functions whereas the third is given in terms of a functional of a 3-dimensional Bessel bridge. The expressions are used for approximating the density. 1Research supported by RiskLab, Switzerland, funded by Credit Suisse Group, Swiss Re and UBS AG. The third author was supported by a Steno grant from the Danish Natural Science Research Council.

Law of the absorption time of some positive self-similar Markov processes

Let X be a spectrally negative self-similar Markov process with 0 as an absorbing state. In this paper, we show that the distribution of the absorption time is absolutely continuous with an infinitely continuously differentiable density. We provide a power series and a contour integral representation of this density. Then, by means of probabilistic arguments, we deduce some interesting analytical properties satisfied by these functions, which include, for instance, several types of hypergeometric functions. We also give several characterizations of the Kesten’s constant appearing in the study of the asymptotic tail distribution of the absorbtion time. We end the paper by detailing some known and new examples. In particular, we offer an alternative proof of the recent result obtained by Bernyk, Dalang and Peskir [Ann. Probab. 36 (2008) 1777–1789] regarding the law of the maximum of spectrally positive Levy stable processes.

A Wiener-Hopf type factorization for the exponential functional of Lévy processes

For a Levy process =( t) t≥0 drifting to -∞, we define the so-called exponential functional as follows: Under mild conditions on , we show that the following factorization of exponential functionals: holds, where × stands for the product of independent random variables, H - is the descending ladder height process of and Y is a spectrally positive Levy 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 I for a large class of Levy 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 itself of independent interest. 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.

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 and Paulsen. 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.

Cauchy problem of the non-self-adjoint Gauss–Laguerre semigroups and uniform bounds for 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.

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 us 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 nontrivial one.

Option Pricing in a One-Dimensional Affine Term Structure Model via Spectral Representations

Under a mild condition on the branching mechanism, we provide an eigenvalue expansion for the pricing semigroup in a one-dimensional positive affine term structure model. This representation, which...

A Sufficient Condition for Continuous-Time Finite Skip-Free Markov Chains to Have Real Eigenvalues

We provide a sufficient condition for the negative of the infinitesimal generator of a continuous-time finite skip-free Markov chain to have only real and non-negative eigenvalues. The condition includes stochastic monotonicity and certain requirements on the transition rates of the chain. We also give a sample path illustration of Markov chains that satisfy the conditions and its Siegmund dual. We illustrate our result by detailing an example which also reveals that our conditions are not necessary.

Convergence Analysis of the Spectral Expansion of Stable Related Semigroups

The purpose of this note is to carry out a convergence analysis of the spectral representation, derived recently in Patie and Savov (Spectral expansions of non-self-adjoint generalized Laguerre semigroups, submitted, 2015), of some non-self-adjoint Markovian semigroups related to spectrally negative α-stable Levy processes conditioned to stay positive. More specifically, we start by performing an error analysis for the spectral type series expansions. Moreover, these semigroups are closely related to a class of invariant semigroups, whose speed of convergence to equilibrium has been studied in Patie and Savov (Spectral expansions of non-self-adjoint generalized Laguerre semigroups, submitted, 2015). Our second aim is to carry out a numerical analysis on the convergence rate which is illustrated with two examples.