History Path Dependent Optimal Control and Portfolio Valuation and Management

History Path Dependent Optimal Control and Portfolio Valuation and Management 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

History Path Dependent Optimal Control and Portfolio Valuation and Management

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2002 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1020244921138
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

PositivitySpringer Journals

Published: Oct 12, 2004

References

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