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.   Google Scholar

[2]

R. Korn and E. Korn, "Option pricing and portfolio optimization'',, American Mathematical Society, (2001).   Google Scholar

[3]

W. Liu and Z. Lin, Strong approximation for a class of stationary processes, Stoch. Proc. Appl., 119 (2009), 249.   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.   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.   Google Scholar

show all references

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.   Google Scholar

[2]

R. Korn and E. Korn, "Option pricing and portfolio optimization'',, American Mathematical Society, (2001).   Google Scholar

[3]

W. Liu and Z. Lin, Strong approximation for a class of stationary processes, Stoch. Proc. Appl., 119 (2009), 249.   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.   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.   Google Scholar

[1]

Mahir Demir, Suzanne Lenhart. A spatial food chain model for the Black Sea Anchovy, and its optimal fishery. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 155-171. doi: 10.3934/dcdsb.2020373

[2]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

[3]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[4]

Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331

[5]

Jiannan Zhang, Ping Chen, Zhuo Jin, Shuanming Li. Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model. Journal of Industrial & Management Optimization, 2021, 17 (2) : 765-777. doi: 10.3934/jimo.2019133

[6]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022

[7]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[8]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[9]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284

[10]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065

[11]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[12]

Gi-Chan Bae, Christian Klingenberg, Marlies Pirner, Seok-Bae Yun. BGK model of the multi-species Uehling-Uhlenbeck equation. Kinetic & Related Models, 2021, 14 (1) : 25-44. doi: 10.3934/krm.2020047

[13]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[14]

Claudio Bonanno, Marco Lenci. Pomeau-Manneville maps are global-local mixing. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1051-1069. doi: 10.3934/dcds.2020309

[15]

Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020365

[16]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091

[17]

Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020367

[18]

Bing Liu, Ming Zhou. Robust portfolio selection for individuals: Minimizing the probability of lifetime ruin. Journal of Industrial & Management Optimization, 2021, 17 (2) : 937-952. doi: 10.3934/jimo.2020005

[19]

Junkee Jeon. Finite horizon portfolio selection problems with stochastic borrowing constraints. Journal of Industrial & Management Optimization, 2021, 17 (2) : 733-763. doi: 10.3934/jimo.2019132

[20]

Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial & Management Optimization, 2021, 17 (2) : 695-709. doi: 10.3934/jimo.2019130

 Impact Factor: 

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (0)

[Back to Top]