# American Institute of Mathematical Sciences

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Stochastic optimal control — A concise introduction
doi: 10.3934/mcrf.2020019

## Non-exponential discounting portfolio management with habit formation

 1 School of Insurance, Central University of Finance and Economics, Beijing, 100081, People's Republic of China 2 Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, 999077, People's Republic of China 3 School of Statistics, East China Normal University, Shanghai, 200241, People's Republic of China

* Corresponding author: Liyuan Lin

Received  April 2019 Revised  January 2020 Published  March 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant No. 11771466, 11301559 and 11601157)

This paper studies the portfolio management problem for an individual with a non-exponential discount function and habit formation in finite time. The investor receives a deterministic income, invests in risky assets, buys insurance and consumes continuously. The objective is to maximize the utility of excessive consumption, heritage and terminal wealth. The non-exponential discounting makes the optimal strategy adopted by a naive person time-inconsistent. The equilibrium for a sophisticated person is Nash subgame perfect equilibrium, and the sophisticated person is time-consistent. We calculate the analytical solution for both the naive strategy and equilibrium strategy in the CRRA case and compare the results of the two strategies. By numerical simulation, we find that the sophisticated individual will spend less on consumption and insurance and save more than the naive person. The difference in the strategies of the naive and sophisticated person decreases over time. Furthermore, if an individual of either type is more patient in the future or has a greater tendency toward habit formation, he/she will consume less and buy less insurance, and the degree of inconsistency will also be increased. The sophisticated person's consumption and habit level are initially lower than those of a naive person but are higher in later periods.

Citation: Jingzhen Liu, Liyuan Lin, Ka Fai Cedric Yiu, Jiaqin Wei. Non-exponential discounting portfolio management with habit formation. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020019
##### References:

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##### References:
The impact of $k$ on $\bar{A}$ and $\hat{A}$
The impact of $k$ in the degree of inconsistency
The impact of habit on $\bar{A}$ and $\hat{A}$
The impact of habit on the degree of inconsistency
Strategy for individual with or without habit
Example of $\bar{H}(t)-\hat{H}(t)$ and $\bar{c}(t)-\hat{c}(t)$
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