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On optimal stopping of risk processes with regime switching

Ferenstein E., Pasternak Winiarski A.

Demonstratio Mathematica

2012

Vol. 45, nr 2

435-454

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.

On a random number of disorders

Szajkowski K.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 1

17--45

We register a random sequence which has three segments being the homogeneous Markov processes. Each segment has its own onestep transition probability law and the length of the segment is unknown and random. It means that at two random moments θ1, θ2, where 0 ≤ θ1 ≤ θ2, the source of observation is changed. In effect, the number of homogeneous segments is random. The transition probabilities of each process are known and the a priori distribution of the disorder moments is given. The former research on such a problem has been devoted to various questions concerning the distribution changes. The random number of distributional segments creates new problems in solutions with relation to analysis of the model with deterministic number of segments. Two cases are presented in detail. In the first one the objective is to stop on or between the disorder moments while in the second one our objective is to find the strategy which immediately detects the distribution changes. Both problems are reformulated to optimal stopping of the observed sequences. The detailed analysis of the problem is presented to show the form of optimal decision function.