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Bayesian learning for the Markowitz portfolio selection problem

Carmine De Franco, Johann Nicolle and Huy\^en Pham
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Carmine De Franco: LPSM UMR 8001
Johann Nicolle: LPSM UMR 8001
Huy\^en Pham: LPSM UMR 8001, CREST

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Abstract: We study the Markowitz portfolio selection problem with unknown drift vector in the multidimensional framework. The prior belief on the uncertain expected rate of return is modeled by an arbitrary probability law, and a Bayesian approach from filtering theory is used to learn the posterior distribution about the drift given the observed market data of the assets. The Bayesian Markowitz problem is then embedded into an auxiliary standard control problem that we characterize by a dynamic programming method and prove the existence and uniqueness of a smooth solution to the related semi-linear partial differential equation (PDE). The optimal Markowitz portfolio strategy is explicitly computed in the case of a Gaussian prior distribution. Finally, we measure the quantitative impact of learning, updating the strategy from observed data, compared to non-learning, using a constant drift in an uncertain context, and analyze the sensitivity of the value of information w.r.t. various relevant parameters of our model.

Date: 2018-11
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