Option pricing in a tychastic market model

Pierre Bernhard
INRIA-Sophia Antipolis-Méditerranée, France

We investigate the simple option pricing problem, either for Vanilla or Digital european options. As compared to the work of Saint-Pierre et al., we have a more specific theory - we are unable to extend it to exotic options - with more analytical results. In particular, we obtain a representation theorem for the Value function, both in continuous and discrete trading, yielding a fast algorithm for the discrete trading case, together with a convergence theorem of this discrete trading value toward the continuous one as the step size vanishes.

Singularities in dynamic programming

Piermarco Cannarsa
Università di Roma 2, Tor Vergata, Italy

Singularities are usually regarded as points to keep away from - when referred to the value function of an optimal control problem. This viewpoint, however, could be totally reversed if one thought of all the data that are condensed at a singular point. This talk will be devoted to singularities of solutions to Hamilton-Jacobi equations, the dynamics of their propagation, and their invariance properties.

Stabilizing differential inclusions and PDEs without uniqueness by noise

Tomas Caraballo
Universidad de Sevilla, Spain

We prove that the asymptotic behavior of partial differential inclusions and partial differential equations without uniqueness of solutions can be stabilized by adding some suitable Ito noise as an external perturbation. We show how the theory previously developed for the single-valued case can be successfully applied to handle these set-valued ones. The theory of random dynamical systems is used as an appropriate tool to solve the problem.

Confinement Problems in Individuals - Population Interactions

Rinaldo Colombo
Università degli studi di Brescia, Italy

Various analytical frameworks are able to describe the interaction between agents and a moving population. First, within a pde setting, this presentation describes a well posedness results. Then, a model based on differential inclusions is presented. In this context, recently obtained positive and negative confinement results are discussed.

Hybrid Control: How Set-Valued Analysis Helps Establish Robust Stability Theory

Rafal Goebel
Loyola University, USA

Modern control algorithms, which involve timers or switching and are often implemented with digital components, leads to dynamical systems where some variable evolve continuously and some change instantaneously. Such systems can be conveniently modeled by a combination of tools from continuous-time and discrete-time settings: differential inclusions and difference inclusions, as well as constraints. This combination leads to hybrid inclusions.

The talk will outline a recently established robust asymptotic stability theory for hybrid inclusions. The role of set-valued analysis be underlined, even for systems without set-valued dynamics. Limitations of the current theory and future challenges will be discussed.

On the role of Viability in Model Predictive Control

Lars Grüne
Universität Bayreuth, Germany

Model predictive control (MPC) is a popular control method which synthesizes a control law on an infinite time horizon by repeatedly solving finite horizon optimal control problems. One of the main reasons for the popularity of MPC is its ability to incorporate state and control constraints. However, due to the fact that in each step only a state constrained optimal control problem on a finite time horizon is solved, viability of the resulting trajectories on the infinite time horizon is not a priori guaranteed. In this talk we present a selection of approaches which allow for rigorous viability statements. Both MPC schemes employing additional terminal constraints and schemes without such terminal constraints are addressed.

Differential equations in (possibly) nonlinear spaces: A short excursion

Thomas Lorenz
Goethe University Frankfurt am Main, Germany

Ordinary differential equations play a central role in science. Their theory is known to be extended successfully from the finite-dimensional Euclidean space to so-called evolution equations in Banach spaces by means of strongly continuous semigroups. Some further extensions are partial differential equations (PDEs) and stochastic differential equations.

For many applications, however, it is difficult to specify a suitable normed vector space. Shapes, for example, do not always have an obvious linear structure if any a priori assumptions about regularity (like convexity or smooth boundaries) are to be avoided and thus analytically, they are described merely as nonempty compact subsets of the Euclidean space. Supplied with the classical Hausdorff distance, they form a metric space instead.

This talk focuses on extending (ordinary) differential equations beyond the traditional border of vector spaces - such as metric spaces.

The long-term goal of this extension is an analytical framework for Cauchy problems in which set-valued evolutions can be handled in the same way as parabolic partial differential equations or semilinear evolution equations, for example. The essential advantage of such a general approach is that we can immediately consider well-posed Cauchy problems of systems although their components are of analytically very different types.

In the 1990s, Jean-Pierre Aubin suggested an extension called "mutational equations". They are similar to "quasidifferential equations" introduced by Panasyuk independently. As a key difference from our point of view, Aubin's concept clearly decomposes the full and usually complex problem (related to the fully nonlinear differential equation) into a class of much simpler problems (called transitions) and feedback. This notion applied to problems in function spaces, for example, covers even some functional differential equations directly. His a priori conditions on transitions, however, are too restrictive for applying the original mutational equations to semilinear evolution equations.

In this talk, several examples serve as motivation how to extend that framework suitably. The list consists of both deterministic and stochastic examples. Invariance and some viability problems can then be handled similarly to what is well known in vector spaces.

Stability of Nonlinear Systems with Discontinuous Control Laws

Richard Vinter
Imperial College London, UK

Discontinuous control laws are widely used to stabilize control systems in the presence of large disturbances. These control laws typically give rise to rapid switching of the controls and the state trajectories must be interpreted in some generalized sense. We discuss interpretations of these generalized state trajectories, and describe how the stability of such systems can be analysed by means of viability theory and the construction of multiple, non-smooth Lyapunov functions.