Abstract
We study semi-Markov risk processes which describe a dynamic of summary capital of an insurance company. The theorems on stability, asymptotic and exponential stability of zero state of the processes with probability 1 are proved. We also investigate the optimal stochastic control of the controlled semi-Markov processes. The Bellman equation for semi-Markov risk processes is derived. Analogue of Dynkin formulae and boundary value problem for semi-Markov random evolutions, and properties of the respected stochastic Liapunov functions are used.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Embrechts, P. and Kluppelberg, C. (1993) Some aspects of Insurance Mathematics. Theory Probability and its Applications, 38, No. 2, 374 - 416.
Swishchuk, A.V. (1993) The Analogue of Dynkin formula and boundary value problems for multiplicative operator functionals of Markov processes, Institute of Mathematics, Kiev, Ukraine, Preprint 93. 44, 17 pp.
Swishchuk, A.V. (1994) Control of evolutionary stochastic systems, Proceed. In Transactions of the 12th Prague Conference on information theory, statistical decision functions and random processes, 235 - 238, Academy of Sciences of the Czech Republic, Prague.
Korolyuk, V.S. and Swishchuk, A.V. (1995) Evolution of Systems in Random Media, CRC Press, Boca Raton, USA.
Swishchuk, A.V. (1996) Approximations and ruin probabilities for semi-Markov risk processes, Institute of Mathematics, Kiev.
Swishchuk, A.V. (1997) Random Evolutions and Their Applications, Kluwer Academic Publishers, Dordrecht.
Swishchuk, A.V. and Goncharova, S.Y. (1998) Stability and control of semi-Markov risk processes in insurance mathematics, In Abstracts of Communications of the 22nd European Meeting of Statisticians & 7th Vilnius Conference on Probability Theory & Mathematical Statistics, 429, TEV, Vilnius.
Kushner, H.J. (1967) Stochastic stability and control, Academic Press, New York-London.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Kluwer Academic Publishers
About this chapter
Cite this chapter
Swishchuk, A. (1999). Stochastic Stability and Optimal Control of Semi-Markov Risk Processes in Insurance Mathematics. In: Janssen, J., Limnios, N. (eds) Semi-Markov Models and Applications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3288-6_19
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3288-6_19
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3290-9
Online ISBN: 978-1-4613-3288-6
eBook Packages: Springer Book Archive