×

zbMATH — the first resource for mathematics

Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping. (English) Zbl 1165.60020
In the paper a class of pathwise inequalities of the form \(H(B_t)\geq M_t+F(L_t)\), where \(B_t\) is Brownian motion, \(L_t\) its local time at zero and \(M_t\) a local martingale, is investigated. The concrete nature of the representation makes the inequality useful for a variety of applications. An example of such usage is constructions and optimality results of Vallois’ Skorokhod embeddings [see [P. Vallois, in: Semin. de probabilites XVII, Proc. 1981/82, Lect. Notes Math. 986, 227–239 (1983; Zbl 0512.60071) and Stochastic Processes Appl. 41, No. 1, 117–155 (1992; Zbl 0754.60089)]. The discussion of their financial interpretation in the context of robust pricing and hedging of options written on the local time is presented. Another application of the inequalities, included in the paper, is solution of a class of optimal stopping problems of the form sup\(_{\tau} \mathbb E[F(L_{\tau})-\int _0^{\tau} \beta(B_s)ds]\). The solution is given via a minimal solution to a system of differential equations and thus resembles the maximality principle described by G. Peškir, [Ann. Probab. 26, No. 4, 1614–1640 (1998; Zbl 0935.60025)].

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
60G44 Martingales with continuous parameter
91B28 Finance etc. (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alili, L. and Kyprianou, A. E. (2005). Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Probab. 15 2062-2080. · Zbl 1083.60034
[2] Arnol’d, V. I. (1992). Ordinary Differential Equations . Springer, Berlin. · Zbl 0744.34001
[3] Azéma, J., Gundy, R. F. and Yor, M. (1980). Sur l’intégrabilité uniforme des martingales continues. In Seminar on Probability XIV ( Paris , 1978/1979 ). Lecture Notes in Mathematics 784 53-61. Springer, Berlin. · Zbl 0442.60046
[4] Breeden, D. T. and Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. J. Business 51 621-651.
[5] Brown, H., Hobson, D. and Rogers, L. C. G. (2001). Robust hedging of barrier options. Math. Finance 11 285-314. · Zbl 1047.91024
[6] Carr, P. (2006). From hyper options to options on local time. Unpublished manuscript.
[7] Carr, P. and Jarrow, R. (1990). The stop-loss start-gain paradox and option valuation: A new decomposition into intrinsic and time value. Rev. Financ. Stud. 3 469-492.
[8] Cox, A. and Hobson, D. (2007). A unifying class of Skorokhod embeddings: Connecting the Azéma-Yor and Vallois embeddings. Bernoulli 13 114-130. · Zbl 1148.60063
[9] Dubins, L. E., Shepp, L. A. and Shiryaev, A. N. (1993). Optimal stopping rules and maximal inequalities for Bessel processes. Teor. Veroyatnost. i Primenen. 38 288-330. · Zbl 0807.60040
[10] Hiriart-Urruty, J.-B. and Lemaréchal, C. (2001). Fundamentals of Convex Analysis . Springer, Berlin. · Zbl 0998.49001
[11] Hobson, D. (1998). Robust hedging of the lookback option. Finance and Stochastics 2 329-347. · Zbl 0907.90023
[12] Hobson, D. (2007). Optimal stopping of the maximum process: A converse to the results of Peskir. Stochastics 79 85-102. · Zbl 1128.60029
[13] Jacka, S. D. (1991). Optimal stopping and best constants for Doob-like inequalities. I. The case p =1. Ann. Probab. 19 1798-1821. · Zbl 0796.60050
[14] Monroe, I. (1972). On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43 1293-1311. · Zbl 0267.60050
[15] Obłój, J. (2004). The Skorokhod embedding problem and its offspring. Probab. Surveys 1 321-392. · Zbl 1189.60088
[16] Obłój, J. (2006). A complete characterization of local martingales which are functions of Brownian motion and its supremum. Bernoulli 12 955-969. · Zbl 1130.60050
[17] Obłój, J. (2007). The maximality principle revisited: On certain optimal stopping problems. In Séminaire de Probabilités XL. Lecture Notes in Mathematics 1899 309-328. Springer, Berlin. · Zbl 1126.60030
[18] Obłój, J. and Yor, M. (2004). An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale. Stochastic Process. Appl. 110 83-110. · Zbl 1075.60038
[19] Peskir, G. (1998). Optimal stopping of the maximum process: The maximality principle. Ann. Probab. 26 1614-1640. · Zbl 0935.60025
[20] Revuz, D. and Yor, M. (2005). Continuous Martingales and Brownian Motion , rev. 3rd ed. Springer, Berlin. · Zbl 1087.60040
[21] Seidenverg, E. (1988). A case of confused identity. Financial Analysts J. 44 63-67.
[22] Vallois, P. (1983). Le problème de Skorokhod sur R : Une approche avec le temps local. In Séminaire de Probabilités XVII. Lecture Notes in Mathematics 986 227-239. Springer, Berlin. · Zbl 0512.60071
[23] Vallois, P. (1992). Quelques inégalités avec le temps local en zero du mouvement brownien. Stochastic Process. Appl. 41 117-155. · Zbl 0754.60089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.