# Abstracts

**Local solution method for the problem of enlargement of filtration**

*Shiqi Song (University Evry Val Essonne, France)*

Thursday June 5, 16:30-17:00 | session 9.4 | Stochastic Analysis | room K

We consider a probability space equipped with two filtrations F and G, one contained in the other. The theory of enlargement of filtration is the research about the relationship between the F-semimartingales and the G-semimartingales. It provides particular techniques to do the stochastic calculus. This theory plays an essential role in credit risk modeling. It also constitutes the major instrument to understand the dependence of a market on the change of information and to inspect the non-arbitrage property. There are two well-known and distinct theories called initial enlargement of filtration and the progressive enlargement of filtration. Such a distinction is the natural consequence of the disparity between the techniques dealing with these two cases, and becomes the standard in the applications of market modeling. However, there exists important models which are covered by neither initial enlargement theory nor the progressive enlargement theory, such as the model of the enlargement by future infima of a positive diffusion. In this talk we present a general theory, called local solution method, to deal with the enlargement of filtration, which contains the initial enlargement and the progressive enlargement theories as corollaries. The local solution method is based on the following observation. The problem of the enlargement of filtration can be defined and be studied locally at every point much like the notion of the derivative defined and computed for a function. In the same time, the problem of the enlargement of filtration also is a global problem much like the integration of a function. The local solution method provides techniques to make use of these two aspects of the problem to yield corresponding solutions. Examples will be given to illustrate how the local solution method constitutes an effective and flexible method. Notably the initial enlargement and the progressive enlargement theories will be investigated anew with the local solution method as well as the model of the enlargement by future infima of a positive diffusion.

**Investment timing and financing strategies under collateral constraint**

*Takashi Shibata (Tokyo Metropolitan University, Japan)*

Thursday June 5, 17:00-17:30 | session 9.4 | Stochastic Analysis | room K

This paper examines the optimal investment timing decision problem of a firm constrained to a debt issuance limit determined by collateral value. We show that the investment thresholds have a U-shaped relation with the debt issuance limit constraints, in that they are increasing (decreasing) with the constraint for high (low) debt issuance limit. Debt issuance limit constraints lead to debt holders experiencing low risk and low returns. That is, the more severe the debt issuance limits, the lower the credit spreads and default probabilities. Our theoretical results are consistent with stylized facts and empirical results.

**Density analysis for BSDEs**

*Thibaut Mastrolia (Université Paris-Dauphine, France)*

Thursday June 5, 17:30-18:00 | session 9.4 | Stochastic Analysis | room K

In recent years the field of Backward Stochastic Di fferential Equations (BSDEs) has been a subject of growing interest in stochastic calculus, since its connection to Finance and more generally to stochastic control problems has been made clear. One of the main issues for the applications is to provide a numerical analysis for the solution of a BSDE. This calls for sharp estimates as well as a deep understanding of the regularity of the solution processes Y and Z. In particular, an important question is whether their law admits a density or not. The current literature on the subject only provides existence results for such a density for the process Y in a Lipschitz setting (see Antonelli and Kohatsu), and for the process Z when the generator is linear in z (see Aboura and Bourguin). In the present work, we give sufficient conditions for the existence of a density for the law of both Y and Z in Lipschitz and even quadratic settings, which are of the utmost importance in applications. Our approach relies heavily on Malliavin calculus, and also allows us to obtain sharp tail estimates for the obtained densities, which we then use to improve the rate of convergence of a truncation procedure in the simulation of quadratic BSDEs, which was first presented by Dos Reis and Imkeller.