Title: Stochastic model for the carbon emissions market: well-posedness and
qualitative behavior.

Abstract. The starting point of the talk is a model for carbon emissions market.
In a first part, I will explain how to model the carbon market as a
forward-backard system of stochastic differential equations on a finite horizon.
Then, I will discuss the well-posedness of the stochastic system. Precisely, I
will investigate three questions:

** existence and uniqueness of a solution. Here, I will make a short review of
the strategy to solve a forward-backward SDE. In particular, I will make the
connection with smoothing properties in PDE theory.

** shape of the boundary condition. The carbon market is governed by a Heaviside
boundary condition. I will show that this makes the stochastic system singular
at maturity time. In particular, I will explain that the underlying forward
process develops a Dirac mass at maturity time: this phenomenon stands for a
loss of the Markov property at maturity time.

** behavior before maturity time. To finish with, I will discuss the law of the
forward process just before maturity time. Specifically, I will investigate the
existence of a density for the transition kernel of the forward process. In this
framework, I will give a brief review of the way to get the existence of the
density by using stochastic analysis and/or PDEs.