Title: Generalized McKean-Vlasov stochastic control problems.

Abstract: I will consider McKean-Vlasov stochastic control problems 
where the cost functions and the state dynamics depend upon the joint 
distribution of the controlled state and the control process. First, I 
will provide a suitable version of the Pontryagin stochastic maximum 
principle, showing that, in the present general framework, pointwise 
minimization of the Hamiltonian with respect to the control is not a 
necessary optimality condition. Then I will take a different 
perspective, and present a variational approach to study a weak 
formulation of such control problems, thereby establishing a new 
connection between those and optimal transport problems on path space.

The talk is based on a joint project with J. Backhoff-Veraguas and R. Carmona.

In this talk we present a framework for discretely compounding
interest rates which is based on the forward price process approach.
This approach has a number of advantages, in particular in the current
market environment. Compared to the classical Libor market models, it
allows in a natural way for negative interest rates and has superb
calibration properties even in the presence of persistently low rates.
Moreover, the measure changes along the tenor structure are simplified
significantly. This property makes it an excellent base for a
post-crisis multiple curve setup. Two variants for multiple curve
constructions will be discussed.

As driving processes we use time-inhomogeneous Lévy processes, which
lead to explicit valuation formulas for various interest rate products
using well-known Fourier transform techniques. Based on these formulas
we present calibration results for the two model variants using market
data for caps with Bachelier implied volatilities.

Hilbert's 19th problem asks if minimizers of "natural" variational integrals are smooth. For the past century, this problem inspired fundamental regularity results for elliptic and parabolic PDES. It also led to the construction of several beautiful counterexamples to regularity. The dichotomy of regularity vs. singularity is related to that of single PDE (the scalar case) vs. system of PDEs (the vectorial case), and low dimension vs. high dimension. I will discuss some interesting recent counterexamples to regularity in low-dimensional vectorial cases, and outstanding open problems. Parts of this are joint works with A. Figalli and O. Savin.

We generalise the existence of combinatorial designs to the setting of subset sums in lattices with coordinates indexed by labelled faces of simplicial complexes. This general framework includes the problem of decomposing hypergraphs with extra edge data, such as colours and orders, and so incorporates a wide range of variations on the basic design problem, notably Baranyai-type generalisations, such as resolvable hypergraph designs, large sets of hypergraph designs and decompositions of designs by designs. Our method also gives approximate counting results, which is new for many structures whose existence was previously known, such as high dimensional permutations or Sudoku squares.

Lincoln Wallen is the Former CTO, Dreamworks Animation. All are welcome