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# Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching

• * Corresponding author: Haixiang Yao. Tel.:+86 2037105360
This research was supported by the National Natural Science Foundation of China (Nos. 71871071, 71471045, 71721001), the Natural Science Foundation of Guangdong Province of China (Nos. 2018B030311004, 2017A030313399, 2017A030313397), the Innovation Team Project of Guangdong Colleges and Universities (No. 2016WCXTD012), the Innovative School Project in Higher Education of Guangdong Province of China (No. GWTP-GC-2017-03)
• The present article investigates a continuous-time mean-variance portfolio selection problem with regime-switching under the constraint of no-shorting. The literature along this line is essentially dominated by the Hamilton-Jacobi-Bellman (HJB) equation approach. However, in the presence of switching regimes, a system of HJB equations rather than a single equation need to be tackled concurrently, which might not be solvable in terms of classical solutions, or even not in the weaker viscosity sense as well. Instead, we first introduce a general result on the sign of geometric Brownian motion with jumps, then derive the efficient portfolio and frontier via the maximum principle approach; in particular, we observe, under a mild technical assumption on the initial conditions, that the no-shorting constraint will consistently be satisfied over the whole finite time horizon. Further numerical illustrations will be provided.

Mathematics Subject Classification: Primary: 49L99; Secondary: 60J27.

 Citation:

• Figure 1.  The value of the stochastic process $P(t, \alpha(t))[x(t)+(\lambda-z)H(t, \alpha(t))]$

Figure 2.  A sample path of the efficient portfolio $u^{*} = (u_1, u_2, u_3)'$

Figure 3.  The process $-[x(t)+(\lambda^{*}-z)H(t, \alpha(t))]$

Figure 4.  The corresponding efficient frontier

Figure 5.  The effects of initial market mode $i_0$

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