Abstract
The Markov property of Markov process functionals which are frequently used in economy, finance, engineering and statistic analysis is studied. The conditions to judge Markov property of some important Markov process functionals are presented, the following conclusions are obtained: the multidimensional process with independent increments is a multidimensional Markov process; the functional in the form of path integral of process with independent increments is a Markov process; the surplus process with the doubly stochastic Poisson process is a vector Markov process. The conditions for linear transformation of vector Markov process being still a Markov process are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Shushin A A I. Non-Markovian stochastic Liouville equation and its Markovian representation, Physical Review E, 2003, 67: 061107.
Anderson J. Best rational approximation to Markov functions, J Approx Theory, 1994, 76(2): 219–232.
Satya N M, Bray A J. Large-deviation functions for nonlinear functionals of a Gaussian stationary Markov process, Physical Review E, 2002, 65, 051112.
Balan R M, Ivanoff B G. A Markov property for set-indexed processes, J Theor Probab, 2002, 15: 553–588.
Dynkin E B. Markov representations of stochastic systems, Russian Math Surveys, 1975, 30(1): 65–104.
Balan R M. Q-Markov random probability measures and their posterior distributions, Stochastic Process Appl, 2004, 109: 295–316.
Ledoux J. Linear dynamics for the state vector of Markov chain functions, Adv Appl Probab, 2004, 36(4): 1198–1211.
Glover J. Markov functions, Ann Inst H Poincar’e Probab Statist, 1991, 27: 221–238.
Sharpe M. General Theory of Markov Processes, New York: Academic Press, 1988.
Jasiulewicy H. Probability of ruin with variable premium rate in a Markovian environment, Insurance: Mathematics and Economics, 2001, 29: 291–296.
Grandell J. Aspects of Risk Theory, New York: Springer-Verlag, 1991.
Taylor G C. Probability of ruin with variable premium rate, Scand Actuarial J, 1980(1): 57–76.
Gerber H U, Shiu E S W. Geometric Brownian motion models for assets and liabilities: from pension funding to optimal dividends, North Amer Actuarial J, 2003, 7(3):37–51.
Jean-Pierre F, Papanicolaou G, Ronnie S K. Derivatives in Financial Markets with Stochastic Volatility, Cambridge: Cambridge University Press, 2000.
Grandits P. A Karamata-type theorem and ruin probabilities for an insurer investing proportionally in the stock market, Insurance: Mathematics and Economics, 2004, 34: 297–305.
Gaier J, Grandits P. Ruin Probabilities and investment under interest force in the presence of regularly varying tails, Scand Actuarial J, 2004, 4: 256–278.
Author information
Authors and Affiliations
Additional information
Supported by the National Natural Science Foundation of China (10671197)
Rights and permissions
About this article
Cite this article
Geng, Xm., Li, L. Markov process functionals in finance and insurance. Appl. Math. J. Chin. Univ. 24, 21–26 (2009). https://doi.org/10.1007/s11766-009-1913-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11766-009-1913-x