Abstract
We use in this chapter 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 viable-capture basin of a dynamical game.
Hence, we shall “compute” the guaranteed viable-capture 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 Hamilton-Jacobi-Isaacs 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—Saint-Pierre viability/capturability algorithm applied to this specific case provides both the valuation function and the associated portfolios.
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References
Aubin, J.-P. (1981). Contingent derivatives of set-valued maps and existence of Solutions to nonlinear inclusions and differential inclusions. In: L. Nachbin ed., Advances in Mathematics, Supplementary studies, pp. 160–232.
Aubin, J.-P. (1986). A viability approach to Lyapunov's second method. In: A. Kurzhanski and K. Sigmund (eds.), Dynamical Systems, Lectures Notes in Economics and Math. Systems, 287:31–38. Springer-Verlag.
Aubin, J.-P. (1991). Viability Theory. Birkhäuser, Boston, Basel, Berlin.
Aubin, J.-P. (1997). Dynamic Economic Theory: A Viability Approach. Springer-Verlag.
Aubin, J.-P. (1999). Impulse Differential Inclusions and Hybrid Systems: A Viability Approach. Lecture Notes, University of California at Berkeley.
Aubin, J.-P. (2000a). Lyapunov Functions for Impulse and Hybrid Control Systems. Proceedings of the CDC 2000 Conference.
Aubin, J.-P. (2000b). Optimal impulse control problems and quasi-variational inequalities thirty years later: a viability approach. In: Contrôle optimal et EDP: Innovations et Applications. IOS Press.
Aubin, J.-P., Bayen, A., Bonneuil, N., and Saint-Pierre, P. (2005). Viability, Control and Games: Regulation of Complex Evolutionary Systems Under Uncertainty and Viability Constraints. Springer-Verlag.
Aubin, J.-P. and Catte, F. (2002). Fixed-Point and Algebraic Properties of Viability Kernels and Capture Basins of Sets. Set-Valued Analysis, 10:379–416.
Aubin, J.-P. and Cellina, A. (1984). Differential Inclusions. Springer-Verlag.
Aubin, J.-P. and Da Prato, G. (1995). Stochastic Nagumo's viability theorem. Stochastic Analysis and Applications, 13:1–11.
Aubin, J.-P. and Da Prato, G. (1998). The viability theorem for stochastic differential inclusions. Stochastic Analysis and Applications, 16:1–15
Aubin, J.-P., Da Prato, G., and Frankowska, H. (2000). Stochastic Invariance for differential inclusions. Set-Valued Analysis, 8:181–201.
Aubin, J.-P. and Dordan, O. (1996). Fuzzy Systems, viability theory and toll sets. In: H. Nguyen (ed.), Handbook of Fuzzy Systems, Modeling and Control, pp. 461–488. Kluwer.
Aubin, J.-P. and Doss, H. (2001). Characterization of stochastic viability of any nonsmotth set involving its generalized contingent curvature. Stochastic Analysis and Applications, 25:951–981.
Aubin, J.-P. and Frankowska, H. (1990). Set-Valued Analysis. Birkhäuser.
Aubin, J.-P. and Frankowska, H. (1996). The viability kernel algorithm for Computing value functions of infinite horizon optimal control problems. J. Math. Anal. Appl, 201:555–576.
Aubin, J.-P. and Haddad, G. (2001a). Cadenced runs of impulse and hybrid control Systems. International Journal Robust and Nonlinear Control, 11:401–415.
Aubin, J.-P. and Haddad, G. (2001b). Path-dependent impulse and hybrid Systems. In: Di Benedetto and Sangiovanni-Vincentelli (eds.), Hybrid Systems: Computation and Control, Proceedings of the HSCC 2001 Conference, LNCS 2034, pp. 119–132. Springer-Verlag.
Aubin, J.-P. and Haddad, G. (2002). History (path) dependent optimal control and portfolio valuation and management. Journla of Positivity, 6:331–358.
Aubin, J.-P., Lygeros, J., Quincampoix, M., Sastry, S., and Seube, N. (2002). Impulse differential inclusions: A viability approach to hybrid Systems. IEEE Transactions on Automatic Control, 47:2–20.
Bardi, M. and Capuzzo Dolcetta, I. (1998). Optimal Control and Viscosity Solutions to Hamilton-Jacobi-Bellman Equations. Birkhäuser.
Bardi, M. and Goatin, P. (1997). A Dirichlet Type Problem for Nonlinear Degenerate Elliptic Equations Arinsing in Time-Optimal Stochastic Control. Preprint SISSA 50/97.
Bardi, M. and Goatin, P. (1998). Invariant sets for controlled degenerate diffusions: a viscosity solution approach. Preprint Padova 2/98.
Bensoussan, A. and Lions, J.-L. (1982). Contrôle impulsionnel et inéquations quasi-variationnelles. Dunod, Paris. (English translation: (1984) Impulse Control and Quasi-Variational Inequalities, Gauthier-Villars).
Bensoussan, A. and Menaldi (1997). Hybrid control and dynamic programming. Dynamics of Continuous, Discrete and Impulse Systems, 3:395–442.
Bernhard, P. (2000a). Max-Plus algebra and mathematical fear in dynamic optimization. J. Set-Valued Analysis.
Bernhard, P. (2000b). Une approche déterministe de l'évaluation des options. In: Optimal Control and Partial Differential Equations, IOS Press.
Bernhard, P. (2000c). A Robust Control Approach to Option Pricing. Cambridge University Press.
Bernhard, P. (2002). Robust control approach to Option pricing, including transaction costs. Annals of Dynamic Games
Bjork, T. (1998). Arbitrage theory in continuous time. Oxford University Press.
Buckdahn, R., Quincampoix, M., and Rascanu, A. (1997). Propriétés de viabilité pour des équations différentielles stochastiques rétrogrades et applications à des équations auxdérivées partielles. Comptes-Rendus de l'Académie des Sciences, 235:1159–1162.
Buckdahn, R., Quincampoix, M., and Rascanu, A. (1998a). Stochastic Viability for Backward Stochastic Differential Equations and Applications to Partial Differential Equations, Un. Bretagne Occidentale, 01-1998.
Buckdahn, R., Peng, S., Quincampoix, M., and Rainer, C. (1998b). Existence of stochastic control under State constraints. Comptes-Rendus de l'Académie des Sciences, 327:17–22.
Buckdahn, R., Cardaliaguet, P., and Quincampoix, M. (2000). A representation formula for the mean curvature motion, UBO 08-2000.
Cardaliaguet, P. (1994). Domaines dicriminants en jeux différentiels. Thèse de l'Université de Paris-Dauphine.
Cardaliaguet, P. (1996). A differential game with two players and one target. SIAM J. Control and Optimization, 34(4):1441–1460.
Cardaliaguet, P. (2000). Introduction à la théorie des jeux différentiels. Lecture Notes, Université Paris-Dauphine.
Cardaliaguet, P., Quincampoix, M., and Saint-Pierre, P. (1999). Setvalued numerical methods for optimal control and differential games. In: Stochastic and differential games. Theory and numerical methods, Annals of the International Society of Dynamical Games, pp. 177–247. Birkhäuser.
Cox, J. and Rubinstein, M. (1985). Options Market. Prentice Hall.
Da Prato, G. and Frankowska, H. (1994). A stochastic Filippov theorem. Stochastic Calculus, 12:409–426.
Da Prato, G. and Frankowska, H. (2001). Stochastic viability for compact sets in terms of the distance function. Dynamics Systems Appl., 10:177–184.
Da Prato, G. and Frankowska, H. (2004). Invariance of stochastic control Systems with deterministic arguments. J. Differential Equations, 200:18–52.
Doss, H. (1977). Liens entre équations différentielles stochastiques et ordinaires. Annales de l'Institute Henri Poincaré, Calcul des Probabilités et Statistique, 23:99–125.
Filipovic, D. (1999). Invariant manifolds for weak Solutions to stochastic equations. Preprint.
Frankowska, H. (1989). Optimal trajectories associated to a solution of contingent Hamilton-Jacobi equations. Applied Mathematics and Optimization, 19:291–311.
Frankowska, H. (1989). Hamilton-Jacobi equation: viscosity Solutions and generalized gradients. J. of Math. Analysis and Appl. 141:21–26.
Frankowska, H. (1991). Lower semicontinuous Solutions to Hamilton-Jacobi-Bellman equations. In: Proceedings of 30th CDC Conference, IEEE, Brighton.
Frankowska, H. (1993). Lower semicontinuous Solutions of Hamilton-Jacobi-Bellman equation. SIAM J. on Control and Optimization.
Gautier, S. and Thibault, L. (1993). Viability for constrained stochastic differential equations. Differential Integral Equations, 6:1395–1414.
Haddad, G. (1981a). Monotone trajectories of differential inclusions with memory. Isr. J. Math., 39:83–100.
Haddad, G. (1981b). Monotone viable trajectories for functional differential inclusions. J. Diff. Eq., 42:1–24.
Haddad, G. (1981c). Topological properties of the set of Solutions for functional differential differential inclusions. Nonlinear Anal. Theory, Meth. Appl., 5:1349–1366.
Jachimiak, W. (1996). A note on invariance for semilinear differential equations. Bull. Pol. Acad. Sc., 44:179–183.
Jachimiak, W. (1998). Stochastic invariance in infinite dimension. Preprint.
Kisielewicz, M. (1995). Viability theorem for stochastic inclusions. Discussiones Mathematicae, Differential Inclusions, 15:61–74.
Milian, A. (1995). Stocastic viability and a comparison theorem. Colloq. Math., 68:297–316.
Milian, A. (1997). Invariance for stochastic equations with regular coefficients. Stochastic Anal. Appl., 15:91–101.
Milian, A. (1998). Comparison theorems for stochastic evolution equations. Preprint.
Olsder, G.J. (1999). Control-theoretic thoughts on option pricing. Preprint.
Pujal, D. (2000). Valuation et gestion dynamiques de portefeuilles, Thèse de l'Université de Paris-Dauphine.
Pujal, D. and Saint-Pierre, P. (2001). L'algorithme du bassin de capture appliqué pour évaluer des options européennes, américaines ou exotiques. Preprint.
Quincampoix, M. (1992). Differential inclusions and target problems. SIAM J. Control and Optimization, 30:324–335.
Rockafellar, R.T. and Wets, R. (1997). Variational Analysis. Springer-Verlag.
Runggaldier, W.J. (2000). Adaptive and robust control peocedures for risk minimization under uncertainty. In: Optimal Control and Partial Differential Equations, pp. 511–520. IOS Press.
Saint-Pierre, P. (1994). Approximation of the viability kernel. Applied Mathematics and Optimisation, 29:187–209.
Saint-Pierre, P. (2005). Approximation of capture basins for hybrid systems. Preprint.
Soner, H.M., Shreve, S.E., and Cvitanic, J. (1995). There is no trivial hedging for Option pricing with transaction costs. The Annals of Probability, 5:327–355.
Soner, H.M. and Touzi, N. (1998). Super-replication under Gamma constraints. SIAM J. Control and Opt., 39:73–96.
Soner, H.M. and Touzi, N. (2000). Dynamic programming for a class of control problems. Preprint.
Soner, H.M. and Touzi, N. (2005). Stochastic target problems, dynamical programming and viscosity Solutions. Preprint.
Tessitore, G. and Zabczyk, J. (1998). Comments on transition semigroups and stochastic invariance. Preprint.
Zabczyk, J. (1973). Optimal control by means of switching. Studia Matematica, 65:161–171.
Zabczyk, J. (1992). Mathematical Control Theory: An Introduction. Birkhäuser.
Zabczyk, J. (1996). Chance and decision: stochastic control in discrete time. Quaderni, Scuola Normale di Pisa.
Zabczyk, J. (1999). Stochastic invariance and consistency of financial models. Preprint Scuola Normale di Pisa.
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Aubin, JP., Pujal, D., Saint-Pierre, P. (2005). Dynamic Management of Portfolios with Transaction Costs under Tychastic Uncertainty. In: Breton, M., Ben-Ameur, H. (eds) Numerical Methods in Finance. Springer, Boston, MA. https://doi.org/10.1007/0-387-25118-9_3
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