A Class of F-Doubly Stochastic Markov Chains

Jecek Jakubowski (University of Warsaw)
Mariusz Andrzej Nieweglowski (Warsaw University of Technology)

Abstract


We define a new class of processes, very useful in applications, $\mathbf{F}$-doubly stochastic Markov chains which contains among others Markov chains. This class is fully characterized by some martingale properties, and one of them is new even in the case of Markov chains. Moreover a predictable representation theorem holds and doubly stochastic property is preserved under natural change of measure.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1743-1771

Publication Date: November 5, 2010

DOI: 10.1214/EJP.v15-815

References

  1. T. Aven. A Theorem for Determining the Compensator of a Counting Process. Scand. J. Statist. 12 (1985), No. 1, 69--72. Math. Review 87f:60074
  2. D. Becherer, M. Schweizer. Classical solutions to reaction-diffusion systems for hedging problems with interacting Itô and point processes. Ann. Appl. Probab. 15 (2005), no. 2, 1111--1144. Math. Review 2005k:60211
  3. T. Bielecki and M. Rutkowski. Credit Risk: Modeling, Valuation and Hedging. Springer Finance. Springer-Verlag Berlin Heidelberg, New York, 2001. Math. Review 2003c:91001
  4. P. Brémaud and M. Yor. Changes of Filtrations and of Probability Measures. Z. Wahrsch. Verw. Gebiete 45 (1978), no. 4, 269--295. Math. Review 80h:60062
  5. R. Cont and P. Tankov Financial modelling with jump processes. Chapman & Hall, 2004. Math. Review 2004m:91004
  6. C. Dellacherie. CapacitÈs et processus stochastiques. Springer, Berlin. (1972). Math. Review 56 #6810
  7. N. El Karoui, S. Peng, and M.C. Quenez. Backward Stochastic Differential Equations in Finance. Math. Finance 7 (1997), no. 1, 1--71. Math. Review 98d:90030
  8. R.D. Gill and S. Johansen. A survey of product-integration with a view toward application in survival analysis. Ann. Statist. 18 (1990), no. 4, 1501--1555. Math. Review 92f:60125
  9. X. Guo and Y. Zeng Intensity Process And Compensator: A New Filtration Expansion Approach And The Jeulin-Yor Theorem. Ann. Appl. Probab. 18 (2008), no. 1, 120--142. Math. Review 2008k:60091
  10. Jacod, Jean and Shiryaev, Albert N. Limit theorems for stochastic processes., Grundlehren der Mathematischen Wissenschaften, 288. Springer-Verlag, Berlin, 1987. Math. Review 2003j:60001
  11. J. Jakubowski and M. Niewęgłowski. Pricing bonds and CDS in the model with rating migration induced by Cox process. Advances in Mathematics of Finance, ed. L. Stettner, Banach Centre Publication. 83 (2008), 159--182. Math. Review 2010j:91193
  12. J. Jakubowski and M. Niewęgłowski. Hedging strategies of payment streams and application to defaultable rating-sensitive claims. To appear in AMaMeF volume Math. Review number not available.
  13. F.C. Klebaner. Introduction to stochastic calculus with applications. Second edition. Imperial College Press, London, 2005. Math. Review 2002h:60001
  14. S. Kusuoka. A remark on default risk models. Adv. Math. Econ. 1 (1999), 69--82. Math. Review 2000i:91060
  15. H. Kunita. Itô's stochastic calculus: Its surprising power for applications Stoch. Proc. Appl. 120 (2010), no. 5, 622--652. Math. Review
  16. D. Lando. On Cox processes and credit risky securities. Rev. Deriv. Res. 2 (1998), 99--120. Math. Review number not available.
  17. G. Last and A. Brandt. Marked Point Processes on the Real Line: The Dynamic Approach. Springer-Verlag Berlin Heidelberg, New York, 1995. Math. Review 97c:60126
  18. E. Pardoux, and S. Peng. Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990), no. 1, 55--61. Math. Review 91e:60171
  19. P. Protter. Stochastic Integration and Differential Equations. Springer-Verlag Berlin Heidelberg, New York, 2004. Math. Review 91i:60148
  20. T. Rolski, H. Schmidli, V. Schmidt and J. Teugels. Stochastic Processes for Insurance and Finance. Wiley Series in Probability and Statistics, 1999. Math. Review 2000a:62273
  21. P. L. Walker. On Lebesgue Integrable Derivatives. Am. Math. Mon. 84 (1977), no. 4, 287--288. Math. Review 55 #5815
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.