Bellman's inclusions and excessive measures

• Autorzy: Zabczyk J.
• Języki publikacji: EN

Abstrakty

EN

The paper is concerned with Bellman's inclusions for the value function of the optimal stopping for a Markov process X on a complete separable metric space E. The author investigates a connection between seemingly unrelated objects; excessive measures, differential inclusions and optimal stopping. Conditions are given under which an evolutionary Bellman inclusion has a strong or weak solution in the Hilbert space L² (E, µ), where p is an excessive measure for X. The solution is identified with the value function of a stopping problem. The stationary Bellman inclusion is treated as well. Specific examples of diffusions with jumps and infinite-dimensional diffusions are discussed. Excessivity of the measure µ plays an essential role in the development. The results are then applied to pricing American options both in finite and infinite dimensions recently investigated by Zhang [32], Mastroeni and Matzeu [20], [21], and Gątarek and Musiela [11].

Słowa kluczowe

EN

Bellman inclusion; Hilbert space; Markov process

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2001
• Tom: Vol. 21, Fasc. 1
• Strony: 101--122
• Opis fizyczny: Bibliogr. 32 poz.

Twórcy

autor

Zabczyk J.

• Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

Bibliografia

• [1] Ph. Benilan, Une remarque sur la convergence des semi-groupes non linéaires, C. R. Acad. Sci. 272 (1971), pp. 1182-1134.
• [2] A. Bensoussan, Stochastic Control by Functional Analysis Methods, North-Holland, Amsterdam 1982.
• [3] A. Bensoussan and J. L. Lions, Applications des inéquations variationnelles en contrôle stochastique, Dunod, Paris 1978.
• [4] P. Billingsley, Convergence of Probability Measures, Wiley, New York 1968.
• [5] H. Brézis, Operateurs maximaux monotones, North-Holland, Amsterdam 1973.
• [6] H. Brézis, M. Crandall and A. Pazy, Perturbations of nonlinear maximal-monotone sets, Comm. Pure Appl. Math. 23 (1970), pp. 123-144,
• [7] P. L. Chow and J. L. Menaldi, Variational inequalities for the control of stochastic partial differential equations, in: Proceedings of the Trento Conference in 1988, Lecture Notes in Math. 1390, pp. 42-52.
• [8] G. Da Prato, Applications croissantes et équations d’évolutions dans les espaces de Banach, Academie Press, 1976.
• [9] N. El Karoui, Les aspects probabilistes du contrôle stochastique, Ecole d’Eté de Probabilités de Saint-Flour IX - 1979, Lecture Notes in Math. 876, Springer, Berlin 1981, pp. 72-238.
• [10] X. Fernique, Régularité des trajectoires des fonctions aléatoires gaussiennes, Lecture Notes in Math. 480 (1975), pp. 2-96.
• [11] D. Gątarek and M. Musiela, Pricing of American interest rate derivatives as optimal stopping of Ornstein-Uhlenbeck processes, manuscript, 1996.
• [12] D. Gątarek and A. Święch, Optimal stopping in Hilbert spaces and pricing of American options, manuscript, 1996.
• [13] R. K. Getoor, Excessive Measures, Birkhäuser, Boston 1990.
• [14] D. Heath, R. Jarrow and A, Morton, Bond pricing and the term structure of interest rates: a new methodology, Econometrica 60 (1992), pp. 77-101.
• [15] P. Jaillet, D. Lamberton and B. Lap eyre, Variational inequalities and the pricing of American options, Acta Appl. Math. 21 (1990), pp. 263-289.
• [16] N. V. Krylov, Controlled Diffusion Processes, Springer, Berlin 1980.
• [17] S. N. Kurtz and T, G. Kurtz, Markov Processes. Characterization and Convergence, Wiley, 1986.
• [18] P. L. Lions, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions, Part III. Uniqueness of viscosity solutions of general second-order equations, J. Funct. Anal. 86 (1989), pp. 1-18.
• [19] Z. M. Ma and M. Roeckner, Introduction of the Theory of (Non-Symmetric) Dirichlet Forms, Springer, 1991.
• [20] L. Mastroeni and M. Matzeu, An integro-differential variational inequality connected with the problem of the American option pricing, Z. Anal. Anwendungen 14 (1995), pp. 869-880.
• [21] L. Mastroeni and M. Matzeu, Stability for the integro-differential variational inequalities of the American option pricing problem, Adv. in Math. Sci. Appl. (to appear).
• [22] M. Musiela, Stochastic PDEs and term structure models, Journées International de Finance, IGR-AFFI, La Baule 1993.
• [23] M. Musiela and M. Rutkowski, Martingale Methods in Finance and Modelling, Springer, 1997.
• [24] R. Myneni, The pricing of American options, Ann. Appl. Probab. 2 (1992), pp. 1-23.
• [25] A. N. Shiryayev, Optimal Stopping Rules, Springer, New York 1978.
• [26] Ł. Stettner and J. Zabczyk,On optimal stopping of Feller processes, Preprint 284, Institute of Mathematics, Polish Academy of Sciences, 1983, pp. 1-30.
• [27] Ł. Stettner and J. Zabczyk, Strong envelopes of stochastic processes and a penalty method, Stochastics 4 (1981), pp. 267-280.
• [28] J. Zabczyk, Optimal stopping on Polish spaces, Preprint 566, Institute of Mathematics, Polish Academy of Sciences, February, 1997.
• [29] J. Zabczyk, Stopping problems on Polish spaces, Ann. Univ. Mariae Curie-Skłodowska, Vol. 51, 1.18 (1997), pp. 181-199.
• [30] J. Zabczyk, Bellman’s inclusions and excessive measures, Preprint No 8, Scuola Normale Superiore di Pisa, March 1998.
• [31] J. Zabczyk, Excessive weights, preprint, Mathematics Institute, University of Warwick, 2001, to appear.
• [32] X. Zhang, Options américaines et modeles des diffusion avec sauts, C. R. Acad. Sci. Paris, Series 1, 317 (1993).