On optimal stopping of risk processes with regime switching

Abstract

In the paper we solve a problem of optimal stopping of a risk process in two alternative settings. We assume that the main characteristics of the risk process change according to unobservable random variable. In the first model we assume that the post-disorder distributions are not known a’priori and are randomly chosen from a finite set of admissible distributions. The second model concentrates on a situation when more than one disorder is possible. For both models optimal stopping rules with respect to given utility function are constructed using dynamic programming methodology.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] N. Bäuerle, U. Rieder, Markov Decision Processes with Applications to Finance, Springer, Berlin 2011.

  • [2] E. Bayraktar, S. Dayanik, I. Karatzas, Adaptive Poisson disorder problem, Ann. Appl. Probab. 16 (2006), 1190–1261.

  • [3] E. Bayraktar, H. V. Poor, Optimal time to change premiums, Math. Methods Oper. Res. 68 (2008), 125–158.

  • [4] T. Bojdecki, J. Hosza, On a generalized disorder problem, Stochastic Process. Appl. 18 (1984), 349–359.

  • [5] F. A. Boshuizen, J. M. Gouweleeuw, General otimal stopping theorems for semi-Markov processes, Adv. in Appl. Probab. 25 (1993), 825–846.

  • [6] P. Brémaud, Point Processes and Queues, Martingale Dynamics, Springer, New York, 1981.

  • [7] M. H. A. Davis, A note on the Poisson disorder problem, in: Mathematical Control Theory, S. Dolecki, C. Olech, J. Zabczyk (eds.), 1976 65–72, Banach Center Publ., Warsaw.

  • [8] M. H. A. Davis, Markov Models and Optimization, Chappman and Hall, 1993.

  • [9] E. Ferenstein, A. Pasternak-Winiarski, Optimal stopping of a risk process with disruption and interest rates, in: Advances in Dynamic Games: Theory, Applications, and Numerical Methods for Differential and Stochastic Games, M. Breton, K. Szajowski (eds.), 2010, 489–508, Birkhäuser, Boston.

  • [10] E. Ferenstein, A. Sierocinski, Optimal stopping of a risk process, Appl. Math. 24 (1997), 335–342.

  • [11] L. I. Galchuk, B. L. Rozovskii, The disorder problem for a Poisson process, Theory Probab. Appl. 16 (1971), 729–734.

  • [12] K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.

  • [13] U. Jensen, An optimal stopping problem in risk theory, Scand. Actuar. J. 1997, 149–159.

  • [14] A. Karpowicz, K. Szajowski, Double optimal stopping of a risk process, Stoch.: Int. J. Probab. Stoch. Process. 79 (2007), 155–167.

  • [15] K. Kuratowski, C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon Sci. 13 (1965), 397–403.

  • [16] B. K. Muciek, Optimal stopping of a risk process: model with interest rates, J. Appl. Probab. 39 (2002), 261–270.

  • [17] B. K. Muciek, K. Szajowski, Optimal stopping of a risk process when claims are covered immediately, in: Mathematical Economics, Toru Maruyama (ed.), RIMS Kyoto University, Kôkyuroku 1557 (2007), 132–139.

  • [18] G. Peskir, A. Shiryayev, Solving the Poisson disorder problem, in: Advances in Finance and Stochastics (2002), 295–312.

  • [19] T. Rolski, H. Schmidli, V. Schmidt, J. Teugels, Stochastic Processes for Insurance and Finance, John Wiley and Sons, Chichester, 1999.

  • [20] W. Sarnowski, K. Szajowski, On-line detection of a part of a sequence with unspecified distribution, Statist. Probab. Lett. 78 (2008), 2511–2516.

  • [21] A. Schöttl, Optimal stopping of a risk reserve process with interest and cost rates, J. Appl. Probab. 35 (1998), 115–123.

  • [22] A. Shiryayev, Optimal Stopping Rules, Springer, New York, 1978.

  • [23] K. Szajowski, A two-disorder detection problem, Appl. Math. 24 (1996), 231–241.

  • [24] K. Szajowski, On a random number of disorders, Probab. Math. Statist. 31 (2011).

  • [25] C. Zhu, Optimal control of the risk process n a regime-switching environment, Automatica 47 (2011), 1570–1579.

OPEN ACCESS

Journal + Issues

The journal publishes research papers in various fields of mathematics, including algebra, analysis, approximation theory, differential equations, mathematical physics, dynamical systems and fractals, discrete mathematics, graph theory, probability theory, functional analysis and statistics.

Search