The Annals of Applied Probability

On the stochastic behaviour of optional processes up to random times

Constantinos Kardaras

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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


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References

  • [1] Barlow, M. T. (1978). Study of a filtration expanded to include an honest time. Z. Wahrsch. Verw. Gebiete 44 307–323.
  • [2] Bichteler, K. (2002). Stochastic Integration with Jumps. Encyclopedia of Mathematics and Its Applications 89. Cambridge Univ. Press, Cambridge.
  • [3] Brémaud, P. and Yor, M. (1978). Changes of filtrations and of probability measures. Z. Wahrsch. Verw. Gebiete 45 269–295.
  • [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.
  • [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.
  • [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.
  • [12] He, S. W., Wang, J. G. and Yan, J. A. (1992). Semimartingale Theory and Stochastic Calculus. Kexue Chubanshe, Beijing.
  • [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.
  • [15] Jeanblanc, M. and Song, S. (2011). An explicit model of default time with given survival probability. Stochastic Process. Appl. 121 1678–1704.
  • [16] Jeulin, T. (1980). Semi-martingales et Grossissement D’une Filtration. Lecture Notes in Math. 833. Springer, Berlin.
  • [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.
  • [18] Jeulin, T. and Yor, M., eds. (1985). Grossissements de Filtrations: Exemples et Applications. Lecture Notes in Math. 1118. Springer, Berlin.
  • [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.
  • [21] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [22] Kardaras, C. (2010). Finitely additive probabilities and the fundamental theorem of asset pricing. In Contemporary Quantitative Finance 19–34. Springer, Berlin.
  • [23] Kardaras, C. (2010). Numéraire-invariant preferences in financial modeling. Ann. Appl. Probab. 20 1697–1728.
  • [24] Kardaras, C. (2014). On the characterisation of honest times that avoid all stopping times. Stochastic Process. Appl. 124 373–384.
  • [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.
  • [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.
  • [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.
  • [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 19701971; Journées Probabilistes de Strasbourg, 1971). Lecture Notes in Math. 258 118–129. Springer, Berlin.
  • [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.
  • [32] Profeta, C., Roynette, B. and Yor, M. (2010). Option Prices as Probabilities: A New Look at Generalized Black–Scholes Formulae. Springer, Berlin.
  • [33] Protter, P. (1990). Stochastic Integration and Differential Equations: A New Approach. Applications of Mathematics (New York) 21. Springer, Berlin.
  • [34] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 1. Cambridge Univ. Press, Cambridge.
  • [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.
  • [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.