Regarding the evolution of financial asset prices governed by an history dependent (path dependent) dynamical system as a prediction mechanism, we provide in this paper the dynamical valuation and management of a portfolio (replicating for instance European, American and other options) depending upon this prediction mechanism (instead of an uncertain evolution of prices, stochastic or tychastic). The problem is actually set in the format of a viability/capturability theory for history dependent control systems and some of their results are then transferred to the specific examples arising in mathematical finance or optimal control. They allow us to provide an explicit formula of the valuation function and to show that it is the solution of a ``Clio Hamilton–Jacobi–Bellman'' equation. For that purpose, we introduce the concept of Clio derivatives of ``history functionals'' in such a way we can give a meaning to such an equation. We then obtain the regulation law governing the evolution of optimal portfolios.
Positivity – Springer Journals
Published: Oct 12, 2004
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