Abstract.
The paper combines two objects rather different at first glance: spaces of stochastic processes having weighted bounded mean oscillation (weighted BMO) and the approximation of certain stochastic integrals, driven by the geometric Brownian motion, by integrals over piece-wise constant integrands. The consideration of the approximation error with respect to weighted BMO implies L p and uniform distributional estimates for the approximation error by a John-Nirenberg type theorem. The general results about weighted BMO are given in the first part of the paper and applied to our approximation problem in the second one.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barlow, M.T., Yor, M.: Semi-martingale inequalities via the Garsia-Rodemich-Rumsey Lemma, and applications to local time. J. Funct. Anal. 49 (2), 198–229 (1982)
Bassily, N.L., Mogyoródi, J.: On the -spaces with general Young function ø. Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 27, 205–214 (1984)
Bonami, A., Lepingle, D.: Fonction maximale et variation quadratique des martingales en présence d’un poids. In: Séminaire de Probabilités XIII, Univ. de Strasbourg. volume 721 of Lect. Notes Math. Springer, 1977/78, pp. 294–306
Burkholder, D.L.: Distribution function inequalities for martingales. Ann. Prob. 1, 19–42 (1973)
Dellacherie, C., Meyer, P.-A.: Probabilities and Potential B. Mathematics Studies 72. North-Holland, 1982
Emery, M.: Une définition faible de BMO. Ann. I.H.P. 21 (1), 59–71 (1985)
Garsia, A.M.: Martingale Inequalities. Seminar Notes on Recent Progress. Benjamin, Reading, 1973
Geiss, S.: In preparation
Geiss, S.: Espaces BMO0,sø et extrapolation. Séminaire d’Initiation à l’Analyse. Univ. Paris 6 120, 1996/97
Geiss, S.: BMOψ-spaces and applications to extrapolation theory. Studia Math. 122, 235–274 (1997)
Geiss, S.: On BMO-spaces of adapted sequences. Séminaire d’Initiation à l’Analyse. Univ. Paris 6 121, 1997/1999
Geiss, S.: On the approximation of stochastic integrals and weighted BMO. In:R. Buckdahn, H.J. Engelbert, and M. Yor, (eds.), Stochastic processes and related topics (Siegmundsburg, 2000), volume 12 of Stochastics Monogr. London, 2002. Taylor & Francis, pp. 133–140
Geiss, S.: Quantitative approximation of certain stochastic integrals. Stochastics and Stochastics Reports 73, 241–270 (2002)
Gobet, E., Temam, E.: Discrete time hedging errors for options with irregular payoffs. Finance and Stochastics 5, 357–367 (2001)
Izumisawa, M., Kazamaki, N.: Weighted norm inequalities for martingales. Tôhoku Math. J. 29, 115–124 (1977)
Kazamaki, N.: Continuous Exponential Martingales and BMO. volume 1579 of Lecture Notes in Mathematics. Springer, 1997
Lenglart, E., Lépingle, D., Pratelli, M.: Présentation unifiée de certaines inégalités de la théorie des martingales. In: Sem. Probab. 14, number 784 in Lecture Notes in Mathematics, Springer, 1980
Lépingle, D.: Quelques inégalités concernant les martingales. Studia Math. 59, 63–83 (1976)
Neveu, J.: Martingales a temps discret. Masson CIE, 1972
Rootzen, H.: Limit distributions for the error in approximations of stochastic integrals. Ann. Prob. 8, 241–251 (1980)
Stein, E.M.: Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, 1993
Strömberg, J.-O.: Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. Indiana Univ. Math. J. 28 (3), 511–544 (1979)
Zhang, R.: Couverture approchée des options Européennes. PhD thesis, Ecole Nationale des Ponts et Chaussées, Paris, 1998
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Geiss, S. Weighted BMO and discrete time hedging within the Black-Scholes model. Probab. Theory Relat. Fields 132, 39–73 (2005). https://doi.org/10.1007/s00440-004-0389-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-004-0389-0