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# Continuous-time mean-variance asset-liability management with stochastic interest rates and inflation risks

• * Corresponding author: zhangck@gdut.edu.cn

This research is supported by National Natural Science Foundation of China (Nos.71571053, 71673061), Natural Science Foundation of Guangdong Province (Nos.2015A030310218, 2016A030313701, 2018A030313687) and Distinguished Innovation Program of Education Commission of Guangdong Province (N0.2015WTSCX014)

• This paper investigates a continuous-time Markowitz mean-variance asset-liability management (ALM) problem under stochastic interest rates and inflation risks. We assume that the company can invest in $n + 1$ assets: one risk-free bond and $n$ risky stocks. The risky stock's price is governed by a geometric Brownian motion (GBM), and the uncontrollable liability follows a Brownian motion with drift, respectively. The correlation between the risky assets and the liability is considered. The objective is to minimize the risk (measured by variance) of the terminal wealth subject to a given expected terminal wealth level. By applying the Lagrange multiplier method and stochastic control approach, we derive the associated Hamilton-Jacobi-Bellman (HJB) equation, which can be converted into six partial differential equations (PDEs). The closed-form solutions for these six PDEs are derived by using the homogenization approach and the variable transformation technique. Then the closed-form expressions for the efficient strategy and efficient frontier are obtained. In addition, a numerical example is presented to illustrate the results.

Mathematics Subject Classification: Primary: 90B50, 90C39; Secondary: 93E20.

 Citation: • • Figure 1.  Impact of $r_0$ on the efficient frontier

Figure 2.  Impact of $\sigma_{r}(t)$ on the efficient frontier

Figure 3.  Impact of $\rho(t)$ on the efficient frontier

Figure 4.  Impact of $I_0$ on the efficient frontier

Figure 5.  Impact of $v(t)$ on the efficient frontier

Figure 6.  Impact of $\eta(t)$ on the efficient frontier

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