Discussion Papers 2021

CIRJE-F-1165	"Portfolio Optimization with Choice of a Probability Measure"
Author Name	Saito, Taiga and Akihiko Takahashi
Date	April 2021
Full Paper	PDF file
Remarks	Revised in December 2021 and March 2022 Published in Proceedings of 2022 IEEE Symposium on Computational Intelligence for Financial Engineering and Economics (CIFEr), May 2022.

Abstract
This paper considers a new problem for portfolio optimization with a choice of a probability measure, partic- ularly optimal investment problem under sentiments. Firstly, we formulate the problem as a sup-sup-inf problem consisting of optimal investment and a choice of a probability measure expressing aggressive and conservative attitudes of the investor. This problem also includes the case where the agent has con- servative and neutral views on risks represented by Brownian motions and degrees of conservativeness differ among the risk. Secondly, we obtain an expression of the volatility process of a backward stochastic differential equation related to the conservative sentiment in order to investigate cases where the sup-sup-inf problem is solved. Specifically, we take a Malliavin calculus approach to solve the problem and obtain an optimal portfolio process. Finally, we provide an expression of the optimal portfolio under the sentiments in two examples with stochastic uncertainties in an exponential utility case and investigate the impact of the sentiments on the portfolio process.
Keywords: Optimal portfolio problem, Uncertainty modeling, Malliavin calculus

Abstract

This paper considers a new problem for portfolio optimization with a choice of a probability measure, partic- ularly optimal investment problem under sentiments. Firstly, we formulate the problem as a sup-sup-inf problem consisting of optimal investment and a choice of a probability measure expressing aggressive and conservative attitudes of the investor. This problem also includes the case where the agent has con- servative and neutral views on risks represented by Brownian motions and degrees of conservativeness differ among the risk. Secondly, we obtain an expression of the volatility process of a backward stochastic differential equation related to the conservative sentiment in order to investigate cases where the sup-sup-inf problem is solved. Specifically, we take a Malliavin calculus approach to solve the problem and obtain an optimal portfolio process. Finally, we provide an expression of the optimal portfolio under the sentiments in two examples with stochastic uncertainties in an exponential utility case and investigate the impact of the sentiments on the portfolio process.

Keywords: Optimal portfolio problem, Uncertainty modeling, Malliavin calculus