L4

(Denis Talay, Inria — joint works with N. Champagnat, N. Perrin, S. Niklitschek Soto)

In this lecture we present recent results on SDEs with weighted local times and discontinuous coefficients. Their solutions allow one to construct probabilistic interpretations of  semilinear PDEs with discontinuous coefficients and transmission boundary conditions in terms of BSDEs which do not satisfy classical conditions.

In this talk we consider optimal stopping problems under a class of coherent risk measures which includes such well known risk measures as weighted AV@R or absolute semi-deviation risk measures. As a matter of fact, the dynamic versions of these risk measures do not have the so-called time-consistency property necessary for the dynamic programming approach. So the standard approaches are not applicable to optimal stopping problems under coherent risk measures. In this paper, we prove a novel representation, which relates the solution of an optimal stopping problem under a coherent risk measure to the sequence of standard optimal stopping problems and hence makes the application of the standard dynamic programming-based approaches possible. In particular, we derive the analogue of the dual representation of Rogers and Haugh and Kogan. Several numerical examples showing the usefulness of the new representation in applications are presented as well.

We investigate the effects of a finite set of agents interacting socially in an equilibrium pricing mechanism. A derivative written on non-tradable underlyings is introduced to the market and priced in an equilibrium framework by agents who assess risk using convex dynamic risk measures expressed by Backward Stochastic Differential Equations (BSDE). An agent is not only exposed to financial and non-financial risk factors, but he also faces performance concerns with respect to the other agents. The equilibrium analysis leads to systems of fully coupled multi-dimensional quadratic BSDEs.

Within our proposed models we prove the existence and uniqueness of an equilibrium. We show that aggregation of risk measures is possible and that a representative agent exists. We analyze the impact of the problem's parameters in the pricing mechanism, in particular how the agent's concern rates affect prices and risk perception.

When firms want to buy back their own shares, they often use the services of investment banks through ASR contracts. ASR contracts are execution contracts including exotic option characteristics (an Asian-type payoff and Bermudian/American exercise dates). In this talk, I will present the different types of ASR contracts usually encountered, and I will present a model in order to (i) price ASR contracts and (ii) find the optimal execution strategy for each type of contract. This model is inspired from the classical (Almgren-Chriss) literature on optimal execution and uses classical ideas from option pricing. It can also be used to price options on illiquid assets. Original numerical methods will be presented.

We will discuss recent results regarding generation and propagation of summability of moments to solution of the Boltzmann equation for variable hard potentials.
These estimates are in direct connection to the understanding of high energy tails and decay rates to equilibrium.

Factorization is a property of global objects that can be built up from local data. In the first half, we introduce the concept of factorization spaces, focusing on two examples relevant for the Geometric Langlands programme: the affine Grassmannian and jet spaces.

In the second half, factorization algebras will be defined including a discussion of how factorization spaces and commutative algebras give rise to examples. Finally, chiral homology is defined as a way to give global invariants of such objects.