arXiv subjects

S. Riedel

Publications and source records attributed to S. Riedel.

3 recordsLinked to original sources

arXiv2024

Invariant Manifolds for Random Parabolic Evolution Equations with almost sectorial operators

In this paper, we develop a way of analyzing the random dynamics of stochastic evolution equations with a non-dense domain. Such problems cover several types of evolution equations. We are particularly interested in evolution equations with non-homogeneous boundary conditions of white noise type. We prove the existence of stable, unstable, and center manifolds around a stationary trajectory by combining integrated semigroup theory and invariant manifold theory. The results are applied to stochastic parabolic equations with white noise at the boundary.

math.PR↗

arXiv2024

Stochastic Control with Signatures

This paper proposes to parameterize open loop controls in stochastic optimal control problems via suitable classes of functionals depending on the driver's path signature, a concept adopted from rough path integration theory. We rigorously prove that these controls are dense in the class of progressively measurable controls and use rough path methods to establish suitable conditions for stability of the controlled dynamics and target functional. These results pave the way for Monte Carlo methods to stochastic optimal control for generic target functionals and dynamics. We discuss the rather versatile numerical algorithms for computing approximately optimal controls and verify their accurateness in benchmark problems from Mathematical Finance.

math.OC↗

arXiv2018

Rough flows

We introduce in this work a concept of rough driver that somehow provides a rough path-like analogue of an enriched object associated with time-dependent vector fields. We use the machinery of approximate flows to build the integration theory of rough drivers and prove well-posedness results for rough differential equations on flows and continuity of the solution flow as a function of the generating rough driver. We show that the theory of semi-martingale stochastic flows developed in the 80's and early 90's fits nicely in this framework, and obtain as a consequence some strong approximation results for general semimartingale flows and provide a fresh look at large deviation theorems for 'Gaussian' stochastic flows.

math.PR↗