A Tychastic Approach to Guaranteed Pricing and Management of Portfolios under Transaction Constraints

Abstract

Dynamic guaranteed pricing and management of a portfolio under transaction constraints is actually a problem straightforwardly set in terms of guaranteed capture basin of a time-dependent target that is viable in a time-dependent environment under (stochastic or tychastic) uncertain systems. The knowledge of the properties of “capture basin” of targets viable in evolving environments under an uncertain evolutionary system can be used for obtaining the corresponding properties for portfolios. They yield at each time both the evaluation of the capital and the transaction rule. They can be computed by viability algorithms and software providing the valuation of optimal portfolio and the management of their evolution. The capital function, which is actually the value function of a differential game, is the solution to a free boundary problem for nonlinear partial differential equations with discontinuous coefficients. This survey provides several examples.