# Time-Consistent Portfolio Selection Under Short-Selling Prohibition: From Discrete to Continuous Setting

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## Abstract

In this paper, we study the time consistent strategies in the mean-variance portfolio selection with short-selling prohibition in both discrete and continuous time settings. Recently, [T. Björk, A. Murgoci, and X. Y. Zhou, Math. Finance, 24 (2014), pp. 1-24] considered the problem with state dependent risk aversion in the sense that the risk aversion is inversely proportional to the current wealth, and they showed that the time consistent control is linear in wealth. Considering the counterpart of their continuous time equilibrium control in the discrete time framework, the corresponding "optimal" wealth process can take negative values; and this negativity in wealth will lead the investor to a risk seeker which results in an unbounded value function that is economically unsound; even more, the limiting of the discrete solutions has shown to be their obtained continuous solution in [T. Björk, A. Murgoci, and X. Y. Zhou, Math. Finance, 24 (2014), pp. 1-24]. To deal this limitation, we eliminate the chance of getting nonpositive wealth by prohibiting short-selling. Using backward induction, the equilibrium control in discrete time setting is shown to be linear in wealth. An application of the extended Hamilton-Jacobi-Bellman equation (see [T. Björk and A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problems, working paper, Stockholm School of Economics, Stockholm, Sweden, 2010]) makes us also conclude that the continuous time equilibrium control is also linear in wealth with investment to wealth ratio satisfying an integral equation uniquely. We also show that the discrete time equilibrium controls converge to that in continuous time setting. Finally, in numerical studies, we illustrate that the constrained strategy in continuous setting can outperform the unconstrained one in some situations as depicted in Figure 8.