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Dynamic management of portfolios with transaction costs under tychastic uncertainty
 In: M. Breton and H. BenAmeur (Eds.). Numerical Methods in Finance
, 2005
"... We use in this paper the viability/capturability approach for studying the problem of dynamic valuation and management of a portfolio with transaction costs in the framework of tychastic control systems (or dynamical games against nature) instead of stochastic control systems. Indeed, the very defin ..."
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Cited by 7 (4 self)
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We use in this paper the viability/capturability approach for studying the problem of dynamic valuation and management of a portfolio with transaction costs in the framework of tychastic control systems (or dynamical games against nature) instead of stochastic control systems. Indeed, the very definition of the guaranteed valuation set can be formulated directly in terms of guaranteed viablecapture basin of a dynamical game. Hence, we shall “compute ” the guaranteed viablecapture basin and find a formula for the valuation function involving an underlying criterion, use the tangential properties of such basins for proving that the valuation function is a solution to HamiltonJacobiIsaacs partial differential equations. We then derive a dynamical feedback providing an adjustment law regulating the evolution of the portfolios obeying viability constraints until it achieves the given objective in finite time. We shall show that the Pujal & SaintPierre viability/capturability algorithm applied to this specific case provides both the valuation function and the associated portfolios. Acknowledgments The authors thank Giuseppe Da Prato, Francine Catté, Halim Doss, Hélène Frankowska, Georges Haddad, Nisard Touzi and Jerzy Zabczyk for many useful discussions and Michèle Breton and Georges Zaccour for inviting us in June 2004 to present these results in the Montréal’s GERAD (Groupe d’études et de recherche en analyse des décisions). Outline The first section is an introduction stating the problem and describing the main results presented. It is intended to readers who are not interested in the mathematical technicalities of the viability approach to financial dynamic valuation and management problems. The second section outlines the viability/capturability strategy and provides the minimal definitions and results of viability theory for deriving in the third and last section sketches the proofs of the main results 1Work supported in part by the European Community’s Human Potential Programme under contract HPRNCT
BoundaryValue Problems for Systems of HamiltonJacobiBellman Inclusions with Constraints
 SIAM J. Control
"... We study in this paper boundaryvalue problems for systems of HamiltonJacobiBellman firstorder partial differential equations and variational inequalities, the solutions of which are constrained to obey viability constraints. They are motivated by some control problems (such as impulse control) ..."
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Cited by 6 (2 self)
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We study in this paper boundaryvalue problems for systems of HamiltonJacobiBellman firstorder partial differential equations and variational inequalities, the solutions of which are constrained to obey viability constraints. They are motivated by some control problems (such as impulse control) and financial mathematics. We shall prove the existence and uniqueness of such solutions in the class of closed setvalued maps, by giving a precise meaning to what a solution means in this case. We shall also provide explicit formulas to this problem. When we deal with HamiltonJacobiBellman equations, we obtain the existence and uniqueness of Frankowska contingent episolutions. We shall deduce these results from the fact that the graph of the solution is the viablecapture basin of the graph of the boundaryconditions under an auxiliary system, and then, from their properties and their characterizations proved in [12, Aubin].
Dynamical qualitative analysis of evolutionary systems
 In Hybrid Systems: Computation and Control, LNCS 2289
, 2002
"... Kuipers ’ QSIM algorithm for tracking the monotonicity properties of solutions to differential equations has been revisited by Dordan by placing it in a rigorous mathematical framework. The Dordan QSIM algorithm provides the transition laws from one qualitative cell to the others. We take up this id ..."
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Cited by 5 (1 self)
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Kuipers ’ QSIM algorithm for tracking the monotonicity properties of solutions to differential equations has been revisited by Dordan by placing it in a rigorous mathematical framework. The Dordan QSIM algorithm provides the transition laws from one qualitative cell to the others. We take up this idea and revisit it at the light of recent advances in the field of “hybrid systems ” and, more generally, “impulse differential equations and inclusions”. Let us consider a family of “qualitative cells Q(a) ” indexed by a parameter a ∈ A: We introduce a dynamical system on the discrete set of qualitative states prescribing an order of visit of the qualitative cells and an evolutionary system govening the “continuous ” evolution of a system, such as a control system. The question arises to study and characterize the set of any pairs of qualitative and quantitative initial states from which start at least one order of visit of the qualitative cells and an continuous evolution visiting the qualitative cells in the prescribed order. This paper is devoted to the issues regarding this question using tools of setvalued analysis and viability theory.
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 upo ..."
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Cited by 3 (2 self)
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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 HamiltonJacobiBellman ” 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. Keywords: HamiltonJacobiBellman equations, history dependent control, path dependent control, functional differential inclusion, viability, capturability, portfolio valuation, portfolio management, Clio derivatives, chaining of functions.