Nomura Seminar
|
Fri, 24/05 16:00 |
Harry Zheng (London) |
Nomura Seminar  |
DH 1st floor SR |
| In this paper we prove a weak necessary and sufficient maximum principle for Markov regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle. | |||
|
Fri, 31/05 16:00 |
Ioannis Karatzas (Columbia) |
Nomura Seminar |
DH 1st floor SR |
| In an equity market with stable capital distribution, a capitalization-weighted index of small stocks tends to outperform a capitalization-weighted index of large stocks.} This is a somewhat careful statement of the so-called "size effect", which has been documented empirically and for which several explanations have been advanced over the years. We review the analysis of this phenomenon by Fernholz (2001) who showed that, in the presence of (a suitably defined) stability for the capital structure, this phenomenon can be attributed entirely to portfolio rebalancing effects, and will occur regardless of whether or not small stocks are riskier than their larger brethren. Collision local times play a critical role in this analysis, as they capture the turnover at the various ranks on the capitalization ladder. We shall provide a rather complete study of this phenomenon in the context of a simple model with stable capital distribution, the so-called “Atlas model" studied in Banner et al.(2005). This is a Joint work with Adrian Banner, Robert Fernholz, Vasileios Papathanakos and Phillip Whitman. | |||
|
Fri, 07/06 16:00 |
Nizar Touzi (Ecole Polytechnique (ParisTech)) |
Nomura Seminar |
DH 1st floor SR |
| tba | |||
|
Fri, 14/06 16:00 |
Kathrin Glau (Technical University Munich) |
Nomura Seminar |
DH 1st floor SR |
| Advanced models such as Lévy models require advanced numerical methods for developing efficient pricing algorithms. Here we focus on PIDE based methods. There is a large arsenal of numerical methods for solving parabolic equations that arise in this context. Especially Galerkin and Galerkin inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense. We therefore classify Lévy processes according to the solution spaces of the associated parabolic PIDEs. We define the Sobolev index of a Lévy process by a certain growth condition on the symbol. It follows that for Lévy processes with a certain Sobolev index b the corresponding evolution problem has a unique weak solution in the Sobolev-Slobodeckii space with index b/2. We show that this classification applies to a wide range of processes. Examples are the Brownian motion with or without drift, generalised hyperbolic (GH), CGMY and (semi) stable Lévy processes. A comparison of the Sobolev index with the Blumenthal-Getoor index sheds light on the structural implication of the classification. More precisely, we discuss the Sobolev index as an indicator of the smoothness of the distribution and of the variation of the paths of the process. An application to financial models requires in particular to admit pure jump processes as well as unbounded domains of the equation. In order to deal at the same time with the typical payoffs which can arise, the weak formulation of the equation has to be based on exponentially weighted Sobolev-Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options. The talk is based on joint work with Ernst Eberlein. | |||