Loïc Chaumont

Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning

1. Measure theory and probability 2. Independence and conditioning 3. Gaussian variables 4. Distributional computations 5. Convergence of random variables 6. Random processes.

Conditionings and path decompositions for Lévy processes

We first give an interpretation for the conditioning to stay positive (respectively, to die at 0) for a large class of Levy processes starting at x > 0. Next, we specify the laws of the pre-minimum and post-minimum parts of a Levy process conditioned to stay positive. We show that, these parts are independent and have the same law as the process conditioned to die at 0 and the process conditioned to stay positive starting at 0, respectively. Finally, in some special cases, we prove the Skorohod convergence of this family of laws when x goes to 0.

Markovian bridges: weak continuity and pathwise constructions

A Markovian bridge is a probability measure taken from a disintegration of the law of an initial part of the path of a Markov process given its terminal value. As such, Markovian bridges admit a natural parameterization in terms of the state space of the process. In the context of Feller processes with continuous transition densities, we construct by weak convergence considerations the only versions of Markovian bridges which are weakly continuous with respect to their parameter. We use this weakly continuous construction to provide an extension of the strong Markov property in which the flow of time is reversed. In the context of self-similar Feller process, the last result is shown to be useful in the construction of Markovian bridges out of the trajectories of the original process.

On the law of the supremum of Levy processes

We show that the law of the overall supremum

Fluctuation theory and exit systems for positive self-similar Markov processes

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Sur certains processus de Lévy conditionnés à rester positifs

\overline{X}_{t}=\sup_{s\let}X_{s}undefined of a Levy process X, before the deterministic time t is equivalent to the average occupation measure μ+t(dx)=∫t0P(Xs∈dx)ds, whenever 0 is regular for both open halflines (−∞,0) and (0,∞). In this case, P(X¯¯¯¯t∈dx) is absolutely continuous for some (and hence for all) t>0 if and only if the resolvent measure of X is absolutely continuous. We also study the cases where 0 is not regular for both halflines. Then we give absolute continuity criterions for the laws of (gt,X¯¯¯¯t) and (gt,X¯¯¯¯t,Xt), where gt is the time at which the supremum occurs before t. The proofs of these results use an expression of the joint law P(gt∈ds,Xt∈dx,X¯¯¯¯t∈dy) in terms of the entrance law of the excursion measure of the reflected process at the supremum and that of the reflected process at the infimum. As an application, this law is made (partly) explicit in some particular instances.

The Lamperti representation of real-valued self-similar Markov processes

For a positive self-similar Markov process, X, we construct a local time for the random set, Θ, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R, H) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set Θ and the process X sampled on the local time scale. The process (R, H) is described in terms of a ladder process linked to the Levy process associated to X via Lamperti’s transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finite-dimensional convergence of (R, H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012–1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying Levy process oscillates.

Some calculations for doubly perturbed Brownian motion

One introduces first a probability distribution of Levy process with no negative jumps and conditioned to be positive With this distribution the process is decomposed at its minimum value. This result is then applied to describe the excursion measure of the reflecting initial Levy process

Invariance principles for local times at the maximum of random walks and Lévy processes

In this paper, we obtain a Lamperti type representation for real-valued self-similar Markov processes, killed at their hitting time of zero. Namely, we represent real-valued self-similar Markov processes as time changed multiplicative invariant processes. Doing so, we complete Kiu’s work [Stochastic Process. Appl. 10 (1980) 183–191], following some ideas in Chybiryakov [Stochastic Process. Appl. 116 (2006) 857–872] in order to characterize the underlying processes in this representation. We provide some examples where the characteristics of the underlying processes can be computed explicitly.

A new fluctuation identity for Lévy processes and some applications

In the present paper we compute the laws of some functionals of doubly perturbed Brownian motion, which is the solution of the equation Xt=Bt+[alpha] sups[less-than-or-equals, slant]t Xs+[beta] infs[less-than-or-equals, slant]t Xs, where [alpha],[beta]