## The Annals of Applied Probability

### The disorder problem for compound Poisson processes with exponential jumps

Pavel V. Gapeev

#### 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

http://projecteuclid.org/euclid.aoap/1106922334

Digital Object Identifier
doi:10.1214/105051604000000981

Mathematical Reviews number (MathSciNet)
MR2115049

Zentralblatt MATH identifier
1068.60062

#### 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. http://projecteuclid.org/euclid.aoap/1106922334.

#### 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.
• Carlstein, E., Müller, H.-G. and Siegmund, D., eds. (1994). Change-Point Problems. IMS, Hayward, CA.
• 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.
• 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.
• 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.
• 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.
• Sato, K. I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
• 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.
• Shiryaev, A. N. (1978). Optimal Stopping Rules. Springer, Berlin.
• Shiryaev, A. N. (1999). Essentials of Stochastic Finance. World Scientific, Singapore.