arXiv subjects

Rémi Lassalle

Publications and source records attributed to Rémi Lassalle.

7 recordsLinked to original sources

arXiv2023

Some applications of the Tsirelson spectral measures to noise filtering problems

This paper investigates applications of the Tsirelson spectral measures to noise filtering problems within the classical Wiener noise framework. We particularly focus on those among those measures associated to square integrable Radon-Nikodym derivatives of laws of observation signals absolutely continuous with respect to the Wiener measure. Explicit formulas are provided to compute the values of the Tsirelson spectral measure of specific sets of interests. Then, several applications to filtering problems are considered, notably to the so-called innovation problem of filtering.

math.PR↗

arXiv2022

A mass transport approach to the optimization of adapted couplings of real valued random variables

In this work, we investigate an optimization problem over adapted couplings between pairs of real valued random variables, possibly describing random times. We relate those couplings to a specific class of causal transport plans between probabilities on the set of real numbers endowed with a filtration, for which their provide a specific representation, which is motivated by a precise characterization of the corresponding deterministic transport plans. From this, under mild hypothesis, the existence of a solution to the problem investigated here is obtained. Furthermore, several examples are provided, within this explicit framework.

math.PR↗

arXiv2022

Average preserving variation processes in view of optimization

In this paper, we investigate specific least action principles for laws of stochastic processes within a framework which stands on filtrations preserving variations. The associated Euler-Lagrange conditions, which we obtain, exhibit a deterministic process in the dynamics aside the canonical martingale term. In particular, taking specific action functionals, extremal processes with respect to those variations encompass specific laws of continuous semi-martingales whose drift characteristic is integrable with independent increments. Then, we relate extremal processes of classical cost functions, in particular of specific entropy functions, to a class of forward-backward systems of Mckean-Vlasov stochastic differential equations.

math.PR↗

arXiv2021

Contraction criteria for Brownian filtrations samplings

This paper investigates the problem to determine whether a given stochastic process generates a sampled Brownian filtration. A fairly general sufficient condition is obtained by applying the Frank H. Clarke contraction criteria to a functional whose construction relies on the quasi-invariance properties of a sampled Wiener measure. The latter are investigated from the precise analytic structure underlying this sampled Brownian motion. In particular, the Cameron-Martin space is identified from usual Lebesgue integrals over a Guseinov measure. As an application we obtain sufficient conditions for the existence of a solution to a class of not necessarily Markovian stochastic differential equations driven by a sampled Brownian motion. By taking a sampling times set which coincides with the unit interval, the results apply to continuous time models.

math.PR↗

arXiv2016

Symmetries and martingales in a stochastic model for the Navier-Stokes equation

A stochastic description of solutions of the Navier-Stokes equation is investigated. These solutions are represented by laws of finite dimensional semi-martingales and characterized by a weak Euler- Lagrange condition. A least action principle, related to the relative entropy, is provided. Within this stochastic framework, by assuming further symmetries, the corresponding invariances are expressed by martingales, stemming from a weak Noether's theorem.

math.PR↗

arXiv2015

Causal transference plans and their Monge-Kantorovich problems

This paper investigates causal optimal transportation problems, in the framework of two Polish spaces, both endowed with filtrations. Specific concretizations yield primal problems equivalent to several classical problems of stochastic control, and of stochastic calculus ; trivial filtrations yield usual problems of optimal transport. Within this framework, primal attainments and dual formulations are obtained, under standard hypothesis, for the related variational problems. These problems are intrinsically related to martingales. Finally, we investigate applications to stochastic frameworks. A straightforward equivalence between specific causal optimization problems, and problems of stochastic control, is obtained. Solutions to a class of stochastic differential equations are characterized, as optimum to specific causal Monge-Kantorovich problems ; the existence of a unique strong solution is related to corresponding Monge problems.

math.PR↗

arXiv2015

Weak calculus of variations for functionals of laws of semi-martingales

We develop a non-anticipating calculus of variations for functionals on a space of laws of continuous semi-martingales, which extends the classical one. We extend Hamilton's least action principle and Noether's theorem to this generalized stochastic framework. As an application we obtain, under mild conditions, a stochastic Euler-Lagrange condition and invariants for the critical points of recent problems in stochastic control, namely for the semi-martingale optimal transportation problems.

math.PR↗