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Speaker: Daria Ghilli
Title: On Neumann problems for nonlocal equations related to jump processes
Abstract: We present some recent results related to the well-posedness of Neumann problems for integro-partial differential equations related to jump
processes. In the classical probabilistic approach to
PDEs, Neumann type boundary conditions are associated to stochastic processes being reflected on
the boundary of the domain. For jump processes which are discontinuous and may exit a domain
without touching its boundary, it turns out that there are many ways to define a "process with a reflection". Also, because of the way the equation and the process are related, defining a reflection
on the boundary will change the equation inside the domain. This is a new nonlocal phenomenon
which is not encountered in the case of continuous processes and PDEs.
Our approach is analytical, i.e. we directly work with the generators and not with the process itself.
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Speaker: Lisandro A. Parente
Title: A generalized monotone control problem with average cost criterion
Abstract: In this work we address the analytical aspects and regularity properties of
an optimal control problem with long run average cost criterion for systems which involves
monotone controls. The approach is carried out through the study of the asymptotic behavior of an associated infinite horizon problem, whose solution is given in terms of the viscosity solutions of an adequate Hamilton-Jacobi-Bellman (HJB) variational inequality. Assuming some controllability properties, these solutions converges to a solution of the original problem HJB system.
We also present a discretization procedure and a numerical scheme, showing some preliminary numerical results.
Joint work with Laura S. Aragone and Pablo A. Lotito