An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise

We study a linear quadratic problem for a system governed by the heat equation on a halfline with Dirichlet boundary control and Dirichlet boundary noise. We show that this problem can be reformulated as a stochastic evolution equation in a certain weighted L2 space. An appropriate choice of weight allows us to prove a stronger regularity for the boundary terms appearing in the infinite dimensional state equation. The direct solution of the Riccati equation related to the associated non-stochastic problem is used to find the solution of the problem in feedback form and to write the value function of the problem.Comment: 16 pages. Many misprints have been correcte

Stochastic Maximum Principle for a PDEs with noise and control on the boundary

In this paper we prove necessary conditions for optimality of a stochastic control problem for a class of stochastic partial differential equations that is controlled through the boundary. This kind of problems can be interpreted as a stochastic control problem for an evolution system in an Hilbert space. The regularity of the solution of the adjoint equation, that is a backward stochastic equation in infinite dimension, plays a crucial role in the formulation of the maximum principle.Comment: 15pg

Solutions and approximations of some Lévy-driven stochastic (partial) differential equations

In this work we look at solutions to stochastic partial differential equations (SPDEs) with noise induced by a Lévy process in the context of Marcus integrals. The canonical Marcus integral is known from the study of SDEs with Lévy noise. We recapture the fundamental results on the existence of solution flows to the Marcus SDE and the convergence of Wong-Zakai approximations. We also prove a generalized Itô formula for said solutions and use this result to establish equations for the inverse flow. We are then looking at extensions of Marcus integrals to the case of SPDEs and find solutions for these equations. Our focus mainly lies on multi-dimensional first-order transport equations driven by Lévy noise. Existence and uniqueness results for the Marcus SPDE are established using a method of characteristics. For second-order equations we prove the existence and uniqueness of mild solutions for equations driven by pure jump Lévy processes, also in terms of Marcus SPDEs. Finally, we study a one-dimensional second-order advection-diffusion equation on the half-line, with Lévy noise at the boundary. Both Dirichlet and Neumann boundary conditions are considered, and the closed form formulae for mild solutions are determined. We also define Wong-Zakai type approximations of the solution by classical solutions and show convergence in the setting of the M1-topology in the Skorokhod space

Elliptic and Parabolic Boundary Value Problems in Weighted Function Spaces

In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted Lq-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt A$_{∞}$-class. In the Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the A$_{p}$-range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators