Markov Switched Difference Equations of Investment Risk Analysis

Proceedings of the 14th Conference on Applied Mathematics APLIMAT 2015 2015
Jevgeņijs Carkovs, Kārlis Šadurskis

This paper deals with linear difference equations with coefficients dependent on the Markov chain. We derive operator equations for the first and the second moments, which can be used by investors for Markowitz approach to portfolio selection. This result is the most convenient for equations with near to constant coefficients. In this case we have succeeded in asymptotic approximation of the above mentioned operator equations by an ordinary difference equations with constant coefficients. Besides we have proved that covariance matrices of solutions can be analyzed as powers of a positive operator in partially ordered Banach space. This permits to formulate the necessary and sufficient mean square stability condition as a spectral problem. The proposed method gives very convenient criterion for ergodic property analysis of GARCH(p,q) processes. Much of our paper is devoted to asymptotic analysis of difference equations subjected to small parameter. For these equation we derive diffusion approximation method and prove that resulting stochastic Ito equation may be used for stationary solution analysis. In order to illustrate of this method availability we analyze the most popular GARCH(1,1) model for portfolio volatility. Our proposal approach permits to develop diffusion approximation for regression model with correlated residuals and to discuss dependence of stationary volatility on correlation coefficient.

Atslēgas vārdi
Stochastic difference equations, GARCH process, Markov dynamical systems,

Carkovs, J., Šadurskis, K. Markov Switched Difference Equations of Investment Risk Analysis. No: Proceedings of the 14th Conference on Applied Mathematics APLIMAT 2015, Slovākija, Bratislava, 3.-5. februāris, 2015. Bratislava: Slovak University of Technology in Bratislava, Publishing House of STU, 2015, 175.-190.lpp. ISBN 978-80-227-4143-3.

Publikācijas valoda
English (en)