arXiv subjects

Larbi Alili

Publications and source records attributed to Larbi Alili.

12 recordsLinked to original sources

arXiv2024

Boundary crossing problems and functional transformations for Ornstein-Uhlenbeck processes

We are interested in the law of the first passage time of an Ornstein-Uhlenbeck process to time-varying thresholds. We show that this problem is connected to the laws of the first passage time of the process to members of a two-parameter family of functional transformations of a time-varying boundary. For specific values of the parameters, these transformations appear in a realisation of a standard Ornstein-Uhlenbeck bridge. We provide three different proofs of this connection. The first one is based on a similar result for Brownian motion, the second uses a generalisation of the so-called Gauss-Markov processes and the third relies on the Lie group symmetry method. We investigate the properties of these transformations and study the algebraic and analytical properties of an involution operator which is used in constructing them. We also show that these transformations map the space of solutions of Sturm-Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we interpret our results through the method of images and give new examples of curves with explicit first passage time densities.

math.PR↗

arXiv2020

On finiteness and tails of perpetuities under a Lamperti-Kiu MAP

Consider a Lamperti-Kiu Markov additive process $(J_t,ξ_t:t\geq0)$ on $\{+,-\}\times\mathbb{R}\cup\infty$ where $J$ is the modulating Markov chain component. First, we study the finiteness of the exponential functional and then consider its moments and tail asymptotics under Cramer's condition. In the strong subexponential case we determine the subexponential tails of the exponential functional under some further assumptions.

math.PR↗

arXiv2019

A comparison of European and Asian options under Markov additive processes

We provide results relating to the integrability, uniform integrability and local integrability of exponential MAPs, which are natural extensions of exponential Levy models. Then, we use Mellin transform and partial integro-differential equation methods to value European options under a such a model. Finally, a comparison is made between the price of a European call option and that of an Asian call option.

math.PR↗

arXiv2019

On Doney's striking factorization of the arc-sine law

R. Doney identifies a striking factorization of the arc-sine law in terms of the suprema of two independent stable processes of the same index by an elegant random walks approximation. In this paper, we provide an alternative proof and a generalization of this factorization based on the theory recently developed for the exponential functional of Lévy processes. As a by-product, we provide some interesting distributional properties for these variables and also some new examples of the factorization of the arc-sine law.

math.PR↗

arXiv2018

Space and time inversions of stochastic processes and Kelvin transform

Let $X$ be a standard Markov process. We prove that a space inversion property of $X$ implies the existence of a Kelvin transform of $X$-harmonic, excessive and operator-harmonic functions and that the inversion property is inherited by Doob $h$-transforms. We determine new classes of processes having space inversion properties amongst transient processes {satisfying the} time inversion property. {For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly.} We treat in details the examples of free scaled power Bessel processes, non-colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.

math.PR↗

arXiv2018

Further studies on square-root boundaries for Bessel processes

We look at decompositions of perpetuities and apply that to the study of the distributions of hitting times of Bessel processes of two types of square root boundaries. These distributions are linked giving a new proof of some Mellin transforms results obtained by David M. DeLong and M. Yor. Several random factorizations and characterizations of the studied distributions are established.

math.PR↗

arXiv2016

Inversion, duality and Doob $h$-transforms for self-similar Markov processes

We show that any $\mathbb{R}^d\setminus\{0\}$-valued self-similar Markov process $X$, with index $α>0$ can be represented as a path transformation of some Markov additive process (MAP) $(θ,ξ)$ in $S_{d-1}\times\mathbb{R}$. This result extends the well known Lamperti transformation. Let us denote by $\widehat{X}$ the self-similar Markov process which is obtained from the MAP $(θ,-ξ)$ through this extended Lamperti transformation. Then we prove that $\widehat{X}$ is in weak duality with $X$, with respect to the measure $π(x/\|x\|)\|x\|^{α-d}dx$, if and only if $(θ,ξ)$ is reversible with respect to the measure $π(ds)dx$, where $π(ds)$ is some $σ$-finite measure on $S_{d-1}$ and $dx$ is the Lebesgue measure on $\mathbb{R}$. Besides, the dual process $\widehat{X}$ has the same law as the inversion $(X_{γ_t}/\|X_{γ_t}\|^2,t\ge0)$ of $X$, where $γ_t$ is the inverse of $t\mapsto\int_0^t\|X\|_s^{-2α}\,ds$. These results allow us to obtain excessive functions for some classes of self-similar Markov processes such as stable Lévy processes.

math.PR↗

arXiv2015

On exponential functionals, harmonic potential measures and undershoots of subordinators

We establish a link between the distribution of an exponential functional I and the undershoots of a subordinator, which is given in terms of the associated harmonic potential measure. This allows us to give a necessary and sufficient condition in terms of the Lévy measure for the exponential functional to be multiplicative infinitely divisible. We then provide a formula for the moment generating function of an exponential functional $I$ and the so called remainder random variable $R$ associated to it. We provide a realization of the remainder random variable $R$ as an infinite product involving independent last position random variables of the subordinator. Some properties of harmonic measures are obtained and some examples are provided.

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↗

arXiv2014

Müntz linear transforms of Brownian motion

We consider a class of linear Volterra transforms of Brownian motion associated to a sequence of Müntz Gaussian spaces and determine explicitly their kernels; some interesting links with Müntz-Legendre polynomials are provided. This gives new explicit examples of progressive Gaussian enlargement of the Brownian filtration. By exploiting a link to stationarity, we give a necessary and sufficient condition for the existence of kernels of infinite order associated to an infinite dimensional Müntz Gaussian space; we also examine when the transformed Brownian motion remains a semimartingale in the filtration of the original process.

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↗

arXiv2008

Further results on some singular linear stochastic differential equations

A class of Volterra transforms, preserving the Wiener measure, with kernels of Goursat type is considered. Such kernels satisfy a self-reproduction property. We provide some results on the inverses of the associated Gramian matrices which lead to a new self-reproduction property. A connection to the classical reproduction property is given. Results are then applied to the study of a class of singular linear stochastic differential equations together with the corresponding decompositions of filtrations. The studied equations are viewed as non-canonical decompositions of some generalized bridges.

math.PR↗