Mean-variance portfolio selection with non-linear wealth dynamics and random coefficients

¹ Shandong university-Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, PR China
² Mathematical Institute and Oxford-Octa Laboratory in Digital Economics, The University of Oxford, Woodstock Road, Oxford OX2 6GG, UK
³ School of Statistics and Mathematics, Shandong University of Finance and Economics, Jinan, Shandong 250100, PR China

^⋆ Corresponding author: shixm@mail.sdu.edu.cn

Received: 19 September 2022
Accepted: 9 April 2024

Abstract

This paper studies the continuous time mean-variance portfolio selection problem with one kind of non-linear wealth dynamics. To deal with the expectation constraint, an auxiliary stochastic control problem is firstly solved by two new generalized stochastic Riccati equations from which a candidate portfolio in feedback form is constructed, and the corresponding wealth process will never cross the vertex of the parabola. In order to verify the optimality of the candidate portfolio, the convex duality (requires the monotonicity of the cost function) is established to give another more direct expression of the terminal wealth level. The variance-optimal martingale measure and the link between the non-linear financial market and the classical linear market are also provided. Finally, we obtain the efficient frontier in closed form. From our results, people are more likely to invest their money in riskless asset compared with the classical linear market.