|2007 Spring School|
Viability: Models, Algorithms and Applications
in Finance and Environmental-Economics
|April 16-20, 2007|
Michèle Breton, CREF, GERAD, HEC Montréal
Georges Zaccour, Chair in game theory and
Chair in game theory and management
The aim of this tutorial is to explain how the mathematical and algorithmic tools of viability theory can be used to solve some issues in both mathematical finance and environmental studies. In a nutshell, viability theory investigates evolutions
in continuous time, discrete time, or an "hybrid" of the two when impulses are involved,
constrained to adapt to an environment,
evolving under contingent, stochastic or tychastic uncertainty,
using for this purpose controls, regulons (regulation controls), subsets of regulons, and in the case of networks, connectionist matrices,
regulated by feedback laws (static or dynamic) that are then "computed" according to given principles, such as the inertia principle, intertemporal optimization, etc.,
co-evolving with their environment (mutational and morphological viability),
and corrected by introducing adequate controls (viability multipliers) when viability or capturability is at stakes.
Main concepts and examples: Viability Kernels of an environment and Capture Basins of a target viable in an environment. Viable Feedbacks. Examples. Viability Algorithms (principle). Cascade of environments. Chaining of feedbacks and concatenation of environments. Tychastic uncertainty. Comparison between stochastic and tychastic viability kernels
Environmental Applications: Inertia Function, dynamic of renewable resources, fisheries, heavy and cyclic evolutions, control of Green House Gases,
Mathematical finance: Viability formulation of the pricing and management of portfolios and cash flows, applications to options, implicit volatility
New perspectives: viability multipliers, impulse systems and mutational equations
Back to mathematics: Viability and Invariance Theorems, Hamilton-Jacobi equations and other issues.
Lecture notes and executables of software will be provided and flexibility is planned to answer questions of interest by the participants (closed-loop pedagogy).