IWS 2015: Statistics and Simulation pp 207-216

# Markowitz Problem for a Case of Random Environment Existence

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 231)

## Abstract

Classical Markowitz model considers n assets with $$R_1, R_2,\ldots ,R_n$$ random profitability and $$r_1, r_2,\ldots ,r_n$$ relevant average, $$\sigma _1^2, \sigma _2^2,\ldots , \sigma _n^2$$ variances, and $$\sigma _{\mu , v}$$, $$\mu , v=1,\ldots ,n$$ covariance. The portfolio is built of these assets, by using weighting coefficients $$\omega _1, \omega _2,\ldots , \omega _n$$, where $$\omega _\mu$$ is the share of asset cost $$\mu$$ in the whole portfolio value. The profitability of such portfolio is a random value $$F(\omega )=\omega _1R_1+ \omega _2R_2+\ldots + \omega _nR_n$$. The cumulative hazard of the portfolio at pre-assigned value of average profitability $$r*$$ can be measured by dispersion $$DF(\omega )$$. It is necessary to determine weighting coefficients by such a way, that minimizes dispersion $$DF(\omega )$$ given assigned value of $$r*$$. A more general supposition considered in this chapter: It is supposed that a random environment exists. The last is described by a continuous-time irreducible Markov chain with k states and known matrix of transition intensities $$\lambda = (\lambda _{i,j})_{k \times k}$$. The reward rate depends on a state of the random environment. For this case, the parameters of Markowitz model are derived.

## Keywords

Markov chain Continuous time Markovic problem

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© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

1. 1.Transport and Telecommunication InstituteRigaLatvia