In this talk, we aim to provide a valuation framework for counterparty credit risk based on a structural default model which incorporates jumps and dependence between the assets of interest. In this framework default is caused by the firm value falling below a prespecified threshold following unforeseeable shocks, which deteriorate its liquidity and ability to meet its liabilities. The presence of dependence between names captures wrong-way risk and right-way risk effects. The structural model traces back to Merton (1974), who considered only the possibility of default occurring at the maturity of the contract; first passage time models starting from the seminal contribution of Black and Cox (1976) extend the original framework to incorporate default events at any time during the lifetime of the contract. However, as the driving risk process used is the Brownian motion, all these models suffers of vanishing credit spreads over the short period - a feature not observed in reality. As a consequence, the Credit Value Adjustment (CVA) would be underestimated for short term deals as well as the so-called gap risk, i.e. the unpredictable loss due to a jump event in the market. Improvements aimed at resolving this issue include for example random default barriers, time dependent volatilities, and jumps. In this contribution, we adopt Lévy processes and capture dependence via a linear combination of two independent Lévy processes representing respectively the systematic risk factor and the idiosyncratic shock. We then apply this framework to the valuation of CVA and DVA related to equity contracts such as forwards and swaps. The main focus is on the impact of correlation between entities on the value of CVA and DVA, with particular attention to wrong-way risk and right-way risk, the inclusion of mitigating clauses such as netting and collateral, and finally the impact of gap risk. Particular attention is also devoted to model calibration to market data, and development of adequate numerical methods for the complexity of the model considered.

Stochastic optimization for sequential decision problems under uncertainty arises in many settings, and as a result as evolved under several canonical frameworks with names such as dynamic programming, stochastic programming, optimal control, robust optimization, and simulation optimization (to name a few).  This is in sharp contrast with the universally accepted canonical frameworks for deterministic math programming (or deterministic optimal control).  We have found that these competing frameworks are actually hiding different classes of policies to solve a single problem which encompasses all of these fields.  In this talk, I provide a canonical framework which, while familiar to some, is not universally used, but should be.  The framework involves solving an objective function which requires searching over a class of policies, a step that can seem like mathematical hand waving.  We then identify four fundamental classes of policies, called policy function approximations (PFAs), cost function approximations (CFAs), policies based on value function approximations (VFAs), and lookahead policies (which themselves come in different flavors).  With the exception of CFAs, these policies have been widely studied under names that make it seem as if they are fundamentally different approaches (policy search, approximate dynamic programming or reinforcement learning, model predictive control, stochastic programming and robust optimization).  We use a simple energy storage problem to demonstrate that minor changes in the nature of the data can produce problems where each of the four classes might work best, or a hybrid.  This exercise supports our claim that any formulation of a sequential decision problem should start with a recognition that we need to search over a space of policies.

We derive a closed form portfolio optimization rule for an investor who is diffident about mean return and volatility estimates, and has a CRRA utility. The novelty is that confidence is here represented using ellipsoidal uncertainty sets for the drift, given a volatility realization. This specification affords a simple and concise analysis, as the optimal portfolio allocation policy is shaped by a rescaled market Sharpe ratio, computed under the worst case volatility. The result is based on a max-min Hamilton-Jacobi-Bellman-Isaacs PDE, which extends the classical Merton problem and reverts to it for an ambiguity-neutral investor.

I will discuss a puzzling theorem about smooth, periodic, real-valued functions on the real line. After introducing the classical Hardy-Littlewood maximal function (which just takes averages over intervals centered at a point), we will prove that if a function has the property that the computation of the maximal function is simple (in the sense that it's enough to check two intervals), then the function is already sin(x) (up to symmetries). I do not know what maximal local averages have to do with the trigonometric function. Differentiation does not help either: the statement equivalently says that a delay differential equation with a solution space of size comparable to C^1(0,1) has only the trigonometric function as periodic solutions.

I will present some recent results concerning the higher gradient integrability of

σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of

div(σ∇u) = 0. When σ is assumed to be symmetric, then the optimal integrability

exponent of the gradient field is known thanks to the work of Astala and Leonetti

& Nesi. I will discuss the case when only the ellipticity is fixed and σ is otherwise

unconstrained and show that the optimal exponent is attained on the class of

two-phase conductivities σ: Ω⊂R27→ {σ1,σ2} ⊂M2×2. The optimal exponent

is established, in the strongest possible way of the existence of so-called

exact solutions, via the exhibition of optimal microgeometries.

(Joint work with V. Nesi and M. Ponsiglione.)

Joint work with Anders Hansen (Cambridge), Olavi Nevalinna (Aalto) and Markus Seidel (Zwickau).             

I will present two recent results concerning the stability of boundary layer asymptotic expansions of solutions of Navier-Stokes with small viscosity. First, we show that the linearization around an arbitrary stationary shear flow admits an unstable eigenfunction with small wave number, when viscosity is sufficiently small. In boundary-layer variables, this yields an exponentially growing sublayer near the boundary and hence instability of the asymptotic expansions, within an arbitrarily small time, in the inviscid limit. On the other hand, we show that the Prandtl asymptotic expansions hold for certain steady flows. Our proof involves delicate construction of approximate solutions (linearized Euler and Prandtl layers) and an introduction of a new positivity estimate for steady Navier-Stokes. This in particular establishes the inviscid limit of steady flows with prescribed boundary data up to order of square root of small viscosity. This is a joint work with Emmanuel Grenier and Yan Guo.

Inspired by a question posed by Lax, in recent years it has received  

an increasing attention the study of quantitative compactness  

estimates for the solution operator $S_t$, $t>0$ that associates to  

every given initial data $u_0$ the corresponding solution $S_t u_0$ of  

a conservation law or of a first order Hamilton-Jacobi equation.



Estimates of this type play a central roles in various areas of  

information theory and statistics as well as of ergodic and learning  

theory. In the present setting, this concept could provide a measure  

of the order of ``resolution'' of a numerical method for the  

corresponding equation.



In this talk we shall first review the results obtained in  

collaboration with O. Glass and K.T. Nguyen, concerning the  

compactness estimates for solutions to conservation laws. Next, we  

shall turn to the  analysis of the Hamilton-Jacobi equation pursued in  

collaboration with P. Cannarsa and K.T.~Nguyen.

In this talk I present a recent result about the free-boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. We study such amplitude equation and prove its nonlinear well-posedness under a stability condition given in terms of a longitudinal strain of the fluid along the discontinuity. This is a joint work with A.Morando and P.Trebeschi.