Geodesic
Approximate optimal stopping of dependent sequences
Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 3, pp. 557-575 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider optimal stopping of sequences of random variables satisfying some asymptotic independence property. Assuming that the embedded planar point processes converge to a Poisson process, we introduce some further conditions to obtain approximation of the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. This limiting problem can be solved in several cases. We apply this method to obtain approximations for the stopping of moving average sequences, of hidden Markov chains, and of max-autoregressive sequences. We also briefly discuss extensions to the case of Poisson cluster processes in the limit.
Keywords: optimal stopping, asymptotic independence, moving average processes, hidden Markov chains.
Mots-clés : Poisson processes 
@article{TVP_2003_48_3_a6,
     author = {R. K\"uhne and L. R\"uschendorf},
     title = {Approximate optimal stopping of dependent sequences},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {557--575},
     year = {2003},
     volume = {48},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2003_48_3_a6/}
}
TY  - JOUR
AU  - R. Kühne
AU  - L. Rüschendorf
TI  - Approximate optimal stopping of dependent sequences
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2003
SP  - 557
EP  - 575
VL  - 48
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TVP_2003_48_3_a6/
LA  - en
ID  - TVP_2003_48_3_a6
ER  - 
%0 Journal Article
%A R. Kühne
%A L. Rüschendorf
%T Approximate optimal stopping of dependent sequences
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2003
%P 557-575
%V 48
%N 3
%U http://geodesic.mathdoc.fr/item/TVP_2003_48_3_a6/
%G en
%F TVP_2003_48_3_a6
R. Kühne; L. Rüschendorf. Approximate optimal stopping of dependent sequences. Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 3, pp. 557-575. http://geodesic.mathdoc.fr/item/TVP_2003_48_3_a6/

[1] Alpuim M. T., Catkan N. A., Hüsler J., “Extremes and clustering of nonstationary max-AR(1) sequences”, Stochastic Process. Appl., 56:1 (1995), 171–184 | DOI | MR | Zbl

[2] Anderson C., “Extreme value theory for a class of discrete distributions with applications to some stochastic processes”, J. Appl. Probab., 7 (1970), 99–113 | DOI | MR | Zbl

[3] Beska M., Klopotowski A., Slominski L., “Limit theorems for random sums of dependent $d$-dimensional random vectors”, Z. Wahrscheinlichkeitstheor. verw. Geb., 61 (1982), 43–57 | DOI | MR | Zbl

[4] Davis R., Resnick S. I., “Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution”, Stochastic Process. Appl., 30:1 (1988), 41–68 | DOI | MR | Zbl

[5] Davis R., Resnick S. I., “Extremes of moving averages of random variables with finite endpoint”, Ann. Probab., 19:1 (1991), 312–328 | DOI | MR | Zbl

[6] Durrett R., Resnick S. I., “Functional limit theorems for dependent variables”, Ann. Probab., 6:5 (1978), 829–846 | DOI | MR | Zbl

[7] Jakubowski A., “Principle of conditioning in limit theorems for sums of random variables”, Ann. Probab., 14:3 (1986), 902–915 | DOI | MR | Zbl

[8] Kennedy D. P., Kertz R. P., “Limit theorems for threshold-stopped random variables with applications to optimal stopping”, Adv. Appl. Probab., 22:2 (1990), 396–411 | DOI | MR | Zbl

[9] Kennedy D. P., Kertz R. P., “The asymptotic behavior of the reward sequence in the optimal stopping of i.i.d. random variables”, Ann. Probab., 19:1 (1991), 329–341 | DOI | MR | Zbl

[10] Kennedy D. P., Kertz R. P., “Limit theorems for suprema, threshold-stopped random variables and last exits of i.i.d. random variables with costs and discounting, with applications to optimal stopping”, Adv. Appl. Probab., 22:2 (1992), 241–266 | DOI | MR

[11] Kühne R., Probleme des asymptotisch optimalen Stoppens, Dissertation, Universität Freiburg, Freiburg, 1997

[12] Kühne R., Rüschendorf L., “Approximation of optimal stopping problems”, Stochastic Process. Appl., 90:2 (2000), 301–325 | DOI | MR | Zbl

[13] Resnick S. I., Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, Berlin, 1987, 320 pp. | MR | Zbl

[14] Rootzen H., “Extreme value theory for moving average processes”, Ann. Probab., 14:2 (1986), 612–652 | DOI | MR | Zbl