Oxford-Man Institute

Duality methods can be very powerful tools for the analysis of stochastic

processes. However, there seems to be no general theory available

yet. In this talk, I will discuss and aim to clarify various notions

of duality, give some recent rather striking examples (applied to

stochastic PDEs, interacting particle systems and combinatorial stochastic

processes)

and try to give some systematic insight into the type of questions

that can in principle be tackled. Finally, I will try to provide you

with some intuition for this fascinating technique.

The talk will explain how a geometric principle gave rise to a new variational description of information-theoretic entropy and how this led to the solution of a problem dating back to the 50's: whether the the central limit theorem is driven by an analogue of the second law of thermodynamics.

This problem is classically addressed by the so-called Skorohod Embedding problem. We instead develop a stochastic control approach. Unlike the previous literature, our formulation seeks the optimal no arbitrage bounds given the knowledge of the distribution at some (or various) point in time. This problem is converted into a classical stochastic control problem by means of convex duality. We obtain a general characterization, and provide explicit optimal bounds in some examples beyond the known classical ones. In particular, we solve completely the case of finitely many given marginals.

In 1986 Kardar, Parisi, and Zhang proposed a stochastic PDE for the motion of driven interfaces,
in particular for growth processes with local updating rules. The solution to the 1D KPZ equation
can be approximated through the weakly asymmetric simple exclusion process. Based on work of 
Tracy and Widom on the PASEP, we obtain an exact formula for the one-point generating function of the KPZ
equation in case of sharp wedge initial data. Our result is valid for all times, but of particular interest is
the long time behavior, related to random matrices, and the finite time corrections. This is joint work with 
Tomohiro Sasamoto.

: Backward error analysis is a technique that has been extremely successful in understanding the behaviour of numerical methods for ordinary differential equations.  It is possible to fit an ODE (the so called modified equation) to a numerical method to very high accuracy. Backward error analysis has been of particular importance in the numerical study of Hamiltonian problems, since it allows to approximate symplectic numerical methods by a perturbed Hamiltonian system, giving an approximate statistical mechanics for symplectic methods. 

Such a systematic theory in the case of numerical methods for stochastic differential equations (SDEs) is currently lacking. In this talk we will describe a general framework for deriving modified equations for SDEs with respect to weak convergence. We will start by quickly recapping of how to derive modified equations in the case of ODEs and describe how these ideas can be generalized in the case of SDEs. Results will be presented for first order methods such as the Euler-Maruyama and the Milstein method. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we will derive a SDE that the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations and in the calculation of effective diffusivities will also be discussed, as well as the use of modified equations  as a tool for constructing higher order methods for stiff stochastic differential equations.

This is joint work with A. Abdulle (EPFL). D. Cohen (Basel), G. Vilmart (EPFL).

Recently, Liang, Lyons and Qian developed a new methodology for the study of backward stochastic differential equations (BSDEs) on general filtered probability spaces. Their approach is based on the analysis of a particular class of functional differential equations, where the driver of the equation does not depend only on the present, but also on the terminal value of the solution.

The purpose of this work is to study fully coupled systems of forward functional differential equations, which are related to a broad class of fully coupled forward-backward stochastic dynamics with respect to general filtrations. In particular, these systems of functional differential equations have a more homogeneous structure with respect to the underlying forward-backward problems, allowing to partly avoid the conflicting nature between the forward and backward components.

Another advantage of the approach is that its generality allows to consider many other types of forward-backward equations not treated in the classical literature: this is shown with the help of several examples, which have interesting applications to mathematical finance and are related to parabolic integro-partial differential equations. In the second part of the talk, we introduce a numerical scheme for the approximation of decoupled systems, based on a time discretization combined with a local iteration approach.

I will introduce the notion of a nonlinear Levy process, discuss basic well-posednes, SDE links and the connection with interacting particles. The talk is aimed to be an introduction to the topic of my recent CUP monograph 'Nonllinear Markov processes and kinetic equations'.

The paper analyses structural models for the evaluation of risky debt following H.E. LELAND [2], with an approach of optimal stopping problem (for instance cf. N. EL KAROUI [1]) and within a more general context: a dividend is paid to equity holders, moreover a different tax schedule is introduced, depending on the firm current value. Actually, an endogenous default boundary is introduced and a nonlinear convex tax schedule allowing for a possible switching in tax benefits. The aim is to find optimal capital structure such that the failure is delayed, meaning how to decrease the failure level VB, anyway preserving D debtholders and E equity holders’interests: for the firm VB is needed as low as possible, for the equity holder, an optimal equity is requested, finally an optimal coupon C is asked  for the total value.

Keywords: corporate debt, optimal capital structure, default,