# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021025

## Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria

 1 School of Mathematical Sciences, Tiangong University, Tianjin 300387, China 2 School of Mathematical Sciences, Tianjin University, Tianjin 300072, China

* Corresponding authors: Hao Chang and Hui Zhao

Received  March 2020 Revised  December 2020 Early access  February 2021

Fund Project: This research is supported by the National Natural Science Foundation of China (Nos.71671122 and 11771329)

This paper studies a robust optimal investment problem under the mean-variance criterion for a defined contribution (DC) pension plan with an ambiguity-averse member (AAM), who worries about model misspecification and aims to find robust optimal strategy. The member has access to a risk-free asset (i.e., cash or bank account) and a risky asset (i.e., the stock) in a financial market. In order to get closer to the actual environment, we assume that both the income level and stock price are driven by Heston's stochastic volatility model. A continuous-time mean-variance model with ambiguity aversion for a DC pension plan is established. By using the Lagrangian multiplier method and stochastic optimal control theory, the closed-form expressions for robust efficient strategy and efficient frontier are derived. In addition, some special cases are derived in detail. Finally, a numerical example is presented to illustrate the effects of model parameters on the robust efficient strategy and the efficient frontier, and some economic implications have been revealed.

Citation: Hao Chang, Jiaao Li, Hui Zhao. Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021025
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##### References:
The effects of volatility parameters $\alpha$, $\sigma _0$, $\lambda _S$ and $\rho$ on $\pi ^{\ast }\left( t \right)$
The effects of income parameters $\mu$, $\sigma _1$ and $k$ on $\pi ^ {\ast }\left( t \right)$
The effect of ambiguity-aversion coefficient $\beta$ on $\pi ^ {\ast }\left( t \right)$
The effects of volatility parameters $\alpha$ and $\sigma _0$ on $\sigma \left( {X\left( T \right)} \right)$
The effects of income parameters $\mu$ and $k$ on $\sigma \left( {X\left( T \right)} \right)$
The effect of ambiguity-aversion coefficient $\beta$ on $\sigma \left( {X\left( T \right)} \right)$
Comparisons of the efficient strategies and the efficient frontiers
When $\rho = -0.65$ and $\lambda _S = 1.5$, we have $\Delta>0$; when $\rho = -0.87$ and $\lambda _S = 12.56$, we get $\Delta = 0$; when $\rho = -0.9$ and $\lambda _S = 10$, we have $\Delta<0$. The effects of different symbols of $\Delta$ on $\pi ^ {\ast }(t)$ and $\sigma (X(T))$
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