Positivity 6: 331–358, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
History Path Dependent Optimal Control and
Portfolio Valuation and Management
and G. HADDAD
14, rue Domat, 75005 Paris Cedex 16, France.
CERMSEM, Maison des Sciences Economiques, Université Paris I,
106–112 Boulevard de l’Hopital, 75645 Paris Cedex 13, France. (E-mail: firstname.lastname@example.org)
(Received 1 September 2001; accepted 25 November 2001)
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 the-
ory 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.
AMS Classiﬁcation: 93C10, 93C15, 93C55, 49J24, 49J40, 49J53
Key words: Hamilton–Jacobi–Bellman equations, history dependent control, path dependent con-
trol, functional differential inclusion, viability, capturability, portfolio valuation, portfolio manage-
ment, Clio derivatives, chaining of functions.
Most models of dynamic valuation and management of portfolios made of shares
of assets, including the replicating portfolios of ﬁnancial options, assume that the
future evolution of prices of the risky assets is uncertain, stochastic or tychastic.
In this paper, we shall assume instead that the future evolution of the asset prices
can be predicted
or forecasted from its history through a convenient prediction
The theory of
(or “robust control”) can be studied in the framework of
dynamical games, when one player plays the role of Nature that chooses (plays) perturbations.
These perturbations, disturbances, parameters that are not under the control of the controller or the
decision-maker, could be called “random variables” if this vocabulary was not already conﬁscated by
probabilists. We suggest to borrow to Charles Peirce the concept of tyche, one of the three words of
classical Greek meaning “chance,” and to call in this case the control system as a
Despite Paul Val
ery’s warning: La pr´evision est un rêve duquel l’´ev´enement nous tire. (Prevision
is a dream from which reality takes us out.)