The Annals of Applied Probability

The disorder problem for compound Poisson processes with exponential jumps

Pavel V. Gapeev

Full-text: Open access

Enhanced PDF (116 KB)

Abstract

The problem of disorder seeks to determine a stopping time which is as close as possible to the unknown time of “disorder” when the observed process changes its probability characteristics. We give a partial answer to this question for some special cases of Lévy processes and present a complete solution of the Bayesian and variational problem for a compound Poisson process with exponential jumps. The method of proof is based on reducing the Bayesian problem to an integro-differential free-boundary problem where, in some cases, the smooth-fit principle breaks down and is replaced by the principle of continuous fit.

Article information

Source
Ann. Appl. Probab. Volume 15, Number 1A (2005), 487-499.

Dates
First available in Project Euclid: 28 January 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1106922334

Digital Object Identifier
doi:10.1214/105051604000000981

Mathematical Reviews number (MathSciNet)
MR2115049

Zentralblatt MATH identifier
1068.60062

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 34K10: Boundary value problems
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 62L15: Optimal stopping [See also 60G40, 91A60] 60J75: Jump processes

Keywords
Disorder (quickest detection) problem Lévy process, compound Poisson process optimal stopping integro-differential free-boundary problem principles of smooth and continuous fit measure of jumps and its compensator Girsanov’s theorem for semimartingales Itô’s formula

Citation

Gapeev, Pavel V. The disorder problem for compound Poisson processes with exponential jumps. Ann. Appl. Probab. 15 (2005), no. 1A, 487--499. doi:10.1214/105051604000000981. https://projecteuclid.org/euclid.aoap/1106922334.


Export citation

References

  • Barndorff-Nielsen, O. E. (1995). Normal inverse Gaussian processes and the modelling of stock returns. Research Report 300, Dept. Theoretical Statistics, Aarhus Univ.
  • Berry, D. A. and Fristedt, B. (1985). Bandit Problems: Sequential Allocation of Experiments. Chapman and Hall, London.
    Mathematical Reviews (MathSciNet): MR813698
    Zentralblatt MATH: 0659.62086
  • Carlstein, E., Müller, H.-G. and Siegmund, D., eds. (1994). Change-Point Problems. IMS, Hayward, CA.
    Mathematical Reviews (MathSciNet): MR1477909
    Zentralblatt MATH: 0942.00037
  • Davis, M. H. A. (1976). A note on the Poisson disorder problem. Banach Center Publ. 1 65--72.
  • Dynkin, E. B. (1963). The optimum choice of the instant for stopping a Markov process. Soviet Math. Dokl. 4 627--629.
    Mathematical Reviews (MathSciNet): MR193670
  • Gal'chuk, L. I. and Rozovskii, B. L. (1971). The ``disorder'' problem for a Poisson process. Theory Probab. Appl. 16 712--716.
  • Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR959133
    Zentralblatt MATH: 0635.60021
  • Kolmogorov, A. N., Prokhorov, Yu. V. and Shiryaev, A. N. (1990). Methods of detecting spontaneously occurring effects. Proc. Steklov Inst. Math. 1 1--21.
  • Mordecki, E. (1999). Optimal stopping for a diffusion with jumps. Finance Stochastics 3 227--236.
    Mathematical Reviews (MathSciNet): MR1810074
    Digital Object Identifier: doi:10.1007/s007800050060
    Zentralblatt MATH: 0926.60034
  • Peskir, G. and Shiryaev, A. N. (2002). Solving the Poisson disorder problem. In Advances in Finance and Stochastics. Essays in Honour of Dieter Sondermann (K. Sandmann and P. Schönbucher, eds.) 295--312. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1929384
    Zentralblatt MATH: 1009.60033
  • Sato, K. I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
    Mathematical Reviews (MathSciNet): MR1739520
  • Shiryaev, A. N. (1961). The detection of spontaneous effects. Soviet Math. Dokl. 2 740--743.
  • Shiryaev, A. N. (1961). The problem of the most rapid detection of a disturbance in a stationary process. Soviet Math. Dokl. 2 795--799.
  • Shiryaev, A. N. (1963). On optimum methods in quickest detection problems. Theory Probab. Appl. 8 22--46.
  • Shiryaev, A. N. (1965). Some exact formulas in a ``disorder'' problem. Theory Probab. Appl. 10 348--354.
  • Shiryaev, A. N. (1967). Two problems of sequential analysis. Cybernetics 3 63--69.
    Mathematical Reviews (MathSciNet): MR272119
  • Shiryaev, A. N. (1978). Optimal Stopping Rules. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR468067
    Zentralblatt MATH: 0391.60002
  • Shiryaev, A. N. (1999). Essentials of Stochastic Finance. World Scientific, Singapore.
    Mathematical Reviews (MathSciNet): MR1695318
    Zentralblatt MATH: 0926.62100

The Institute of Mathematical Statistics

The Annals of Applied Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more