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.
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References
Aubin, J.-P.: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, Advances in Mathematics, Supplementary Studies, Ed. L. Nachbin. 160-232, 1981.
Aubin, J.-P.: Smallest Lyapunov functions of differential inclusions, J. Differential and Integral Equations 2 (1989).
Aubin, J.-P.: Viability Theory, Birkhäuser, Berlin and Boston, 1991.
Aubin, J.-P.: Boundary-value problems for systems of first-order partial differential inclusions, NoDEA 7 (2000), 61-84.
Aubin, J.-P.: Viability kernels and capture basins of sets under differential Inclusions, SIAM J. Control 40 (2001), 853-881.
Aubin, J.-P. and Catte, F.: Fixed-point and algebraic properties of viability kernels and capture basins of sets, to appear, Set-Valued Analysis.
Aubin, J.-P. and Dordan, O.: Impulsive optimal control and stopping time problems in finite horizon (2001).
Aubin, J.-P. and Doss, H.: Dynamic management of portfolios with impulse transactions under tychastic uncertainty, to appear, Stochastics Analysis and Application.
Aubin, J.-P. and Doss, H.: Itô and Stratonovitch stochastic viability, 2001.
Aubin, J.-P., Pujal, D. and Saint-Pierre, P.: Dynamic management of portfolios with transaction costs under tychastic uncertainty, 2001, preprint.
Aubin, J.-P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Boston, and Berlin, 1990.
Aubin, J.-P. and Haddad, G.: Cadenced runs of impulse and hybrid control systems, International Journal Robust and Nonlinear Control, 2001.
Aubin, J.-P. and Haddad, G.: Path-Dependent Impulse and Hybrid Systems, in: Hybrid Systems: Computation and Control, Di Benedetto and E. Sangiovanni-Vincentelli Eds, Di Benedetto & Sangiovanni-Vincentelli Eds, Proceedings of the HSCC 2001 Conference, LNCS 2034, Springer, Berlin, 2001, pp. 119-132.
Aubin, J.-P. and Haddad, G.: Detectability under Impulse differential inclusions, preprint.
Aubin, J.-P. and Haddad, G.: Co-evolution of asset prices and porfolios of shareholders, in preparation.
Bernhard, P.: Une approche déterministe de l'évaluation des options, in Optimal Control and Partial Differential Equations, IOS Press, 2000.
Bernhard, P.: A Robust Control Approach to Option Pricing, Cambridge University Press, Cambridge and New York, 2000.
Bernhard, P.: Robust control approach to option pricing, including transaction costs, Annals of Dynamic Games (2002).
Buckdahn, R., Peng, S., Quincampoix, M. and Rainer, C.: Existence of stochastic control under state constraints, Comptes-Rendus de l'Académie des Sciences 327 (1988), 17-22.
Buckdahn, R., Cardaliaguet, P. and Quincampoix, M.: A representation formula for the mean curvature motion, UBO 08, 2000.
Cardaliaguet, P., Quincampoix, M. and Saint-Pierre, P.: Contribution à l'étude des jeux différentiels quantitatifs et qualitatifs avec contrainte sur l'état, Comptes-Rendus de l'Académie des Sciences 321 (1995), 1543-1548.
Frankowska, H.: L'équation d'Hamilton-Jacobi contingente, Comptes-Rendus de l'Académie des Sciences, PARIS, Série 1 304 (1987), 295-298
Frankowska, H.: Optimal trajectories associated to a solution of contingent Hamilton-Jacobi equations IEEE, 26th, CDC Conference, Los Angeles, CA, December 9-11, 1987.
Frankowska, H.: Optimal trajectories associated to a solution of contingent Hamilton-Jacobi equations, Applied Mathematics and Optimization 19 (1989), 291-311.
Frankowska, H.: Hamilton-Jacobi equation: viscosity solutions and generalized gradients, J. of Math. Analysis and Appl. 141 (1989), 21-26.
Frankowska, H.: Lower semicontinuous solutions to Hamilton-Jacobi-Bellman equations, Proceedings of 30th CDC Conference, IEEE, Brighton, December 11-13, 1991.
Frankowska, H.: Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equation, SIAM J. on Control and Optimization, 1993.
Haddad, G.: Monotone trajectories of differential inclusions with memory, Isr. J. Math. 39 (1981), 83-100.
Haddad, G.: Monotone viable trajectories for functional differential inclusions, J. Diff. Eq. 42 (1981), 1-24.
Haddad, G.: Topological properties of the set of solutions for functional differential differential inclusions, Nonlinear Anal. Theory, Meth. Appl. 5 (1981), 1349-1366.
Peirce, C.: Evolutionary Love, The Monist, 1983.
Pujal, D.: Valuation et Gestion Dynamiques de Portefeuilles, Thèse de l'Université de Paris-Dauphine, 2000.
Pujal, D. and Saint-Pierre, P.: L'algorithme du bassin de capture appliqué pour évaluer des options européennes, américaines ou exotiques, preprint, 2001.
Rockafellar, R. T. and West, R.: Variational Analysis, Springer, Berlin and New York, 1997.
Soner, H. M. and Touzi, N.: Super-replication under gamma constraints, SIAM J. Control and Opt. 39 (1998), 73-96.
Soner, H. M. and Touzi, N.: Dynamic Programming for a Class of Control Problems, 2000.
Soner, H. M. and Touzi, N.: Stochastic target problems, dynamical programming and viscosity solutions, SIAM J. Control. Opt.
Soner, H. M. and Touzi, N.: Dynamic programming for stochastic target problems and geometric flows, to appear, J. of the European Mathematical Society.
Soner, H. M. and Touzi, N.: A stochastic representation for mean curvature type geometric flows, to appear, preprint.
Soner, H. M. and Touzi, N.: Set-valued viscosity solutions and stochastic reachability flows, to appear, preprint.
Zabczyk, J.: Chance and decision: stochastic control in discrete time, Quaderni, Scuola Normale di Pisa, 1996.
Zabczyk, J.: Stochastic invariance and consistency of financial models, Scuola Normale di Pisa, Preprint, 1999.
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Aubin, JP., Haddad, G. History Path Dependent Optimal Control and Portfolio Valuation and Management. Positivity 6, 331–358 (2002). https://doi.org/10.1023/A:1020244921138
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DOI: https://doi.org/10.1023/A:1020244921138