# Mean-variance portfolio selection with nonlinear wealth dynamics and random coefficients

## TL;DR

This paper addresses a continuous-time mean-variance portfolio problem with nonlinear wealth dynamics, deriving explicit solutions and revealing a preference for riskless assets over classical linear market models.

## Contribution

It introduces generalized stochastic Riccati equations to solve the nonlinear control problem and establishes convex duality for optimality verification.

## Key findings

- Explicit closed-form efficient frontier derived.
- Optimal portfolio favors riskless assets more than in linear markets.
- Link established between nonlinear and classical linear market models.

## Abstract

This paper studies the continuous time mean-variance portfolio selection problem with one kind of non-linear wealth dynamics. To deal 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.