# American Institute of Mathematical Sciences

2015, 2015(special): 1041-1049. doi: 10.3934/proc.2015.1041

## Optimal portfolios based on weakly dependent data

 1 Colledge of Science and Technology, Nihon University, Narashino-dai, Funabashi 274-8501 2 Center for Medical Education and Sciences, Yamanashi University, Shimogato, Chuo 409-3898, Japan 3 Department of Mathematics, Tokyo City University, 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo 158-8557 4 Department of Mathematics, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan

Received  September 2014 Revised  March 2015 Published  November 2015

Let $\{\xi_k, k=1,2, \ldots\}$ be a strictly stationary sequence of centered $d$-dimensional random vectors satisfying the strong mixing condition. Using $\{\xi_k\}$, we consider a stochastic difference equation with a random volatility composed by $d$ stocks and a random trend and show a convergence theorem. In the one-dimensional case, the solution of this difference equation converges almost surely to a Black-Scholes type model. The purpose of this paper is to extend the results to multi-dimensional cases. Using the result, we obtain an approximations of $d$ stocks prices models with random volatilities. We also give examples of optimal portfolios for the models.
Citation: Hiroshi Takahashi, Tatsuhiko Saigo, Shuya Kanagawa, Ken-ichi Yoshihara. Optimal portfolios based on weakly dependent data. Conference Publications, 2015, 2015 (special) : 1041-1049. doi: 10.3934/proc.2015.1041
##### References:
 [1] J. Komlòs, P. Major and G. Tusnàdy, An approximation of partial sums of independent RV's and the sample DF.I, Z. Wahrsch. Verw. Gebiete., 32 (1975), 111-131.  Google Scholar [2] R. Korn and E. Korn, "Option pricing and portfolio optimization'', American Mathematical Society, Providence, 2001.  Google Scholar [3] W. Liu and Z. Lin, Strong approximation for a class of stationary processes Stoch. Proc. Appl., 119 (2009), 249-280.  Google Scholar [4] H. Takahashi, S. Kanagawa and K. Yoshihara, Asymptotic behavior of solutions of some difference equations defined by weakly dependent random vectors. Stoch. Anal. Appl., 33 (2015), 740-755. Google Scholar [5] K. Yoshihara, Asymptotic behavior of solutions of Black-Scholes type equations based on weakly dependent random variables, Yokohama Math. J., 58 (2012), 1-15.  Google Scholar

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##### References:
 [1] J. Komlòs, P. Major and G. Tusnàdy, An approximation of partial sums of independent RV's and the sample DF.I, Z. Wahrsch. Verw. Gebiete., 32 (1975), 111-131.  Google Scholar [2] R. Korn and E. Korn, "Option pricing and portfolio optimization'', American Mathematical Society, Providence, 2001.  Google Scholar [3] W. Liu and Z. Lin, Strong approximation for a class of stationary processes Stoch. Proc. Appl., 119 (2009), 249-280.  Google Scholar [4] H. Takahashi, S. Kanagawa and K. Yoshihara, Asymptotic behavior of solutions of some difference equations defined by weakly dependent random vectors. Stoch. Anal. Appl., 33 (2015), 740-755. Google Scholar [5] K. Yoshihara, Asymptotic behavior of solutions of Black-Scholes type equations based on weakly dependent random variables, Yokohama Math. J., 58 (2012), 1-15.  Google Scholar
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