## Journal of Applied Probability

### Optimal sequential change detection for fractional diffusion-type processes

#### Abstract

The problem of detecting an abrupt change in the distribution of an arbitrary, sequentially observed, continuous-path stochastic process is considered and the optimality of the CUSUM test is established with respect to a modified version of Lorden's criterion. We apply this result to the case that a random drift emerges in a fractional Brownian motion and we show that the CUSUM test optimizes Lorden's original criterion when a fractional Brownian motion with Hurst index H adopts a polynomial drift term with exponent H+1/2.

#### Article information

Source
J. Appl. Probab., Volume 50, Number 1 (2013), 29-41.

Dates
First available in Project Euclid: 20 March 2013

https://projecteuclid.org/euclid.jap/1363784422

Digital Object Identifier
doi:10.1239/jap/1363784422

Mathematical Reviews number (MathSciNet)
MR3076770

Zentralblatt MATH identifier
1349.62368

#### Citation

Chronopoulou, Alexandra; Fellouris, Georgios. Optimal sequential change detection for fractional diffusion-type processes. J. Appl. Probab. 50 (2013), no. 1, 29--41. doi:10.1239/jap/1363784422. https://projecteuclid.org/euclid.jap/1363784422

#### References

• Basseville, M. and Nikiforov, I. V. (1993). Detection of Abrupt Changes: Theory and Application. Prentice-Hall, Engelwood Cliffs, NJ.
• Beibel, M. (1996). A note on Ritov's Bayes approach to the minimax property of the cusum procedure. Ann Statist. 24, 1804–1812.
• Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.
• Cheridito, P., Kawaguchi, H. and Maejima, M. (2003). Fractional Ornstein–Uhlenbeck processes. Electron. J. Prob. 8, 14pp.
• Chronopoulou, A. and Viens, F. G. (2012). Estimation and pricing under long-memory stochastic volatility. Ann. Finance 8, 379–403.
• Comte, F. and Renault, E. (1998). Long memory in continuous-time stochastic volatility models. Math. Finance 8, 291–323.
• Decreusefond, L. and Üstünel, A. S. (1999). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10, 177–214.
• Hadjiliadis, O. and Poor, H. V. (2008). Quickest Detection. Cambridge University Press.
• Hawkins, D. M. and Olwell, D. H. (1998). Cumulative Sum Charts and Charting for Quality Improvement. Springer, New York.
• Hu, Y., Nualart, D. and Song, J. (2009). Fractional martingales and characterization of the fractional Brownian motion. Ann. Prob. 37, 2404–2430.
• Hurst, H. (1951). Long term storage capacity of reservoirs. Trans. Amer. Soc. Civil Eng. 116, 770–799.
• Kleptsyna, M. L. and Le Breton, A. (2002). Statistical analysis of the fractional Ornstein–Uhlenbeck type process. Statist. Infer. Stoch. Process. 5, 229–248.
• Kleptsyna, M. L., Le Breton, A. and Roubaud, M.-C. (2000). Parameter estimation and optimal filtering for fractional type stochastic systems. Statist. Infer. Stoch. Process. 3, 173–182.
• Lorden, G. (1971). Procedures for reacting to a change in distribution. Ann. Math. Statist. 42, 1897–1908.
• Mishura, J. and Valkeila, E. (2011). An extension of the Lévy characterization to fractional Brownian motion. Ann. Prob. 39, 439–470.
• Molchan, G. (1969). Gaussian processes which are asymptotically equivalent to a power of $\lambda$. Theory Prob. Appl. 14, 530–532.
• Moustakides, G. V. (1986). Optimal stopping times for detecting changes in distributions. Ann. Statist. 14, 1379–1387.
• Moustakides, G. V. (2004). Optimality of the CUSUM procedure in continuous time. Ann. Statist. 32, 302–315.
• Moustakides, G. V. (2008). Sequential change detection revisited. Ann. Statist. 2, 787–807.
• Norros, I., Valkeila, E. and Virtamo, J. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5, 571–587.
• Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin.
• Page, E. S. (1954). Continuous inspection schemes. Biometrika 41, 100–115.
• Pollak, M. (1985). Optimal detection of a change in distribution. Ann. Statist. 13, 206–227.
• Prakasa Rao, B. L. S. (2004). Sequential estimation for fractional Ornstein–Uhlenbeck type process. Sequent. Anal. 23, 33–44.
• Prakasa Rao, B. L. S. (2005). Sequential testing for simple hypotheses for processes driven by fractional Brownian motion. Sequent. Anal. 24, 189–203.
• Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.
• Ritov, Y. (1990). Decision theoretic optimality of the CUSUM procedure. Ann. Statist. 18, 1464–1469.
• Rogers, L. C. G. (1997). Arbitrage with fractional Brownian motion. Math. Finance 7, 95–105.
• Shiryaev, A. N. (1978). Optimal Stopping Rules. Springer, New York.
• Shiryaev, A. N. (1996). Minimax optimality of the method of cumulative sums (CUSUM) in the continuous time case. Russian Math. Surveys 51, 750–751.
• Tudor, C. A. and Viens, F. G. (2007). Statistical aspects of the fractional stochastic calculus. Ann. Statist. 35, 1183–1212.
• Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282.