The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 25, Number 2 (2015), 429-464.
On the stochastic behaviour of optional processes up to random times
Full-text: Open access
Abstract
In this paper, a study of random times on filtered probability spaces is undertaken. The main message is that, as long as distributional properties of optional processes up to the random time are involved, there is no loss of generality in assuming that the random time is actually a randomised stopping time. This perspective has advantages in both the theoretical and practical study of optional processes up to random times. Applications are given to financial mathematics, as well as to the study of the stochastic behaviour of Brownian motion with drift up to its time of overall maximum as well as up to last-passage times over finite intervals. Furthermore, a novel proof of the Jeulin–Yor decomposition formula via Girsanov’s theorem is provided.
Article information
Source
Ann. Appl. Probab., Volume 25, Number 2 (2015), 429-464.
Dates
First available in Project Euclid: 19 February 2015
Permanent link to this document
https://projecteuclid.org/euclid.aoap/1424355119
Digital Object Identifier
doi:10.1214/13-AAP976
Mathematical Reviews number (MathSciNet)
MR3313744
Zentralblatt MATH identifier
1316.60057
Subjects
Primary: 60G07: General theory of processes 60G44: Martingales with continuous parameter
Keywords
Random times randomised stopping times times of maximum last passage times
Citation
Kardaras, Constantinos. On the stochastic behaviour of optional processes up to random times. Ann. Appl. Probab. 25 (2015), no. 2, 429--464. doi:10.1214/13-AAP976. https://projecteuclid.org/euclid.aoap/1424355119
References
- [1] Barlow, M. T. (1978). Study of a filtration expanded to include an honest time. Z. Wahrsch. Verw. Gebiete 44 307–323.Mathematical Reviews (MathSciNet): MR509204
- [2] Bichteler, K. (2002). Stochastic Integration with Jumps. Encyclopedia of Mathematics and Its Applications 89. Cambridge Univ. Press, Cambridge.Mathematical Reviews (MathSciNet): MR1906715
- [3] Brémaud, P. and Yor, M. (1978). Changes of filtrations and of probability measures. Z. Wahrsch. Verw. Gebiete 45 269–295.Mathematical Reviews (MathSciNet): MR511775
- [4] Delbaen, F. and Schachermayer, W. (1995). Arbitrage possibilities in Bessel processes and their relations to local martingales. Probab. Theory Related Fields 102 357–366.
- [5] Delbaen, F. and Shirakawa, H. (2002). No arbitrage condition for positive diffusion price processes. Asia-Pacific Financial Markets 9 159–168.
- [6] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge.Mathematical Reviews (MathSciNet): MR2722836
- [7] Elliott, R. J., Jeanblanc, M. and Yor, M. (2000). On models of default risk. Math. Finance 10 179–195.
- [8] Elworthy, K. D., Li, X. M. and Yor, M. (1997). On the tails of the supremum and the quadratic variation of strictly local martingales. In Séminaire de Probabilités, XXXI. Lecture Notes in Math. 1655 113–125. Springer, Berlin.
- [9] Fernholz, E. R. (2002). Stochastic Portfolio Theory: Stochastic Modelling and Applied Probability. Applications of Mathematics (New York) 48. Springer, New York.Mathematical Reviews (MathSciNet): MR1894767
- [10] Föllmer, H. (1972). The exit measure of a supermartingale. Z. Wahrsch. Verw. Gebiete 21 154–166.
- [11] Guo, X. and Zeng, Y. (2008). Intensity process and compensator: A new filtration expansion approach and the Jeulin–Yor theorem. Ann. Appl. Probab. 18 120–142.Mathematical Reviews (MathSciNet): MR2380894
Digital Object Identifier: doi:10.1214/07-AAP447
Project Euclid: euclid.aoap/1199890018 - [12] He, S. W., Wang, J. G. and Yan, J. A. (1992). Semimartingale Theory and Stochastic Calculus. Kexue Chubanshe, Beijing.Mathematical Reviews (MathSciNet): MR1219534
- [13] Jacod, J. and Shiryaev, A. N. (1998). Local martingales and the fundamental asset pricing theorems in the discrete-time case. Finance Stoch. 2 259–273.
- [14] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin.Mathematical Reviews (MathSciNet): MR1943877
- [15] Jeanblanc, M. and Song, S. (2011). An explicit model of default time with given survival probability. Stochastic Process. Appl. 121 1678–1704.Mathematical Reviews (MathSciNet): MR2811019
Digital Object Identifier: doi:10.1016/j.spa.2011.04.002 - [16] Jeulin, T. (1980). Semi-martingales et Grossissement D’une Filtration. Lecture Notes in Math. 833. Springer, Berlin.Mathematical Reviews (MathSciNet): MR604176
- [17] Jeulin, T. and Yor, M. (1978). Grossissement d’une filtration et semi-martingales: Formules explicites. In Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977). Lecture Notes in Math. 649 78–97. Springer, Berlin.Mathematical Reviews (MathSciNet): MR519998
- [18] Jeulin, T. and Yor, M., eds. (1985). Grossissements de Filtrations: Exemples et Applications. Lecture Notes in Math. 1118. Springer, Berlin.Mathematical Reviews (MathSciNet): MR884713
- [19] Kallsen, J. (2003). $\sigma$-localization and $\sigma$-martingales. Teor. Veroyatn. Primen. 48 177–188.
- [20] Karatzas, I. and Kardaras, C. (2007). The numéraire portfolio in semimartingale financial models. Finance Stoch. 11 447–493.Mathematical Reviews (MathSciNet): MR2335830
Digital Object Identifier: doi:10.1007/s00780-007-0047-3 - [21] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.Mathematical Reviews (MathSciNet): MR1121940
- [22] Kardaras, C. (2010). Finitely additive probabilities and the fundamental theorem of asset pricing. In Contemporary Quantitative Finance 19–34. Springer, Berlin.Mathematical Reviews (MathSciNet): MR2732838
Digital Object Identifier: doi:10.1007/978-3-642-03479-4_2 - [23] Kardaras, C. (2010). Numéraire-invariant preferences in financial modeling. Ann. Appl. Probab. 20 1697–1728.Mathematical Reviews (MathSciNet): MR2724418
Digital Object Identifier: doi:10.1214/09-AAP669
Project Euclid: euclid.aoap/1282747398 - [24] Kardaras, C. (2014). On the characterisation of honest times that avoid all stopping times. Stochastic Process. Appl. 124 373–384.Mathematical Reviews (MathSciNet): MR3131298
Digital Object Identifier: doi:10.1016/j.spa.2013.07.012 - [25] Kardaras, C. (2014). A time before which insiders would not undertake risk. In Inspired by Finance (the Musiela Festschrift) (Y. Kabanov, M. Rutkowski and T. Zariphopoulou, eds.) 349–362. Springer, Cham.Mathematical Reviews (MathSciNet): MR3204224
Digital Object Identifier: doi:10.1007/978-3-319-02069-3_16 - [26] Kramkov, D. and Sîrbu, M. (2006). On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 16 1352–1384.Mathematical Reviews (MathSciNet): MR2260066
Digital Object Identifier: doi:10.1214/105051606000000259
Project Euclid: euclid.aoap/1159804984 - [27] Kusuoka, S. (1999). A remark on default risk models. In Advances in Mathematical Economics, Vol. 1 (Tokyo, 1997). Adv. Math. Econ. 1 69–82. Springer, Tokyo.Mathematical Reviews (MathSciNet): MR1722700
Digital Object Identifier: doi:10.1007/978-4-431-65895-5_5 - [28] Lando, D. (1998). On Cox processes and credit risky securities. Review of Derivatives Research 2 610–612.
- [29] Meyer, P. A. (1972). La mesure de H. Föllmer en théorie des surmartingales. In Séminaire de Probabilités, VI (Univ. Strasbourg, Année Universitaire 1970–1971; Journées Probabilistes de Strasbourg, 1971). Lecture Notes in Math. 258 118–129. Springer, Berlin.Mathematical Reviews (MathSciNet): MR368131
- [30] Nikeghbali, A. and Yor, M. (2006). Doob’s maximal identity, multiplicative decompositions and enlargements of filtrations. Illinois J. Math. 50 791–814 (electronic).
- [31] Parthasarathy, K. R. (2005). Probability Measures on Metric Spaces. AMS Chelsea Publishing, Providence, RI.Mathematical Reviews (MathSciNet): MR2169627
- [32] Profeta, C., Roynette, B. and Yor, M. (2010). Option Prices as Probabilities: A New Look at Generalized Black–Scholes Formulae. Springer, Berlin.Mathematical Reviews (MathSciNet): MR2582990
- [33] Protter, P. (1990). Stochastic Integration and Differential Equations: A New Approach. Applications of Mathematics (New York) 21. Springer, Berlin.Mathematical Reviews (MathSciNet): MR1037262
- [34] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 1. Cambridge Univ. Press, Cambridge.Mathematical Reviews (MathSciNet): MR1796539
- [35] Shiryaev, A. N. and Cherny, A. S. (2000). Some distributional properties of a Brownian motion with a drift and an extension of P. Lévy’s theorem. Theory Probab. Appl. 44 412–418.
- [36] Tsirelson, B. (1998). Within and beyond the reach of Brownian innovation. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998) 311–320 (electronic). Doc. Math., Extra Vol. III.Mathematical Reviews (MathSciNet): MR1648165
- [37] Yor, M. (1978). Grossissement d’une filtration et semi-martingales: Théorèmes généraux. In Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977). Lecture Notes in Math. 649 61–69. Springer, Berlin.Mathematical Reviews (MathSciNet): MR519996
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