Singular and impulse stochastic control problems motivated by optimal harvesting and portfolio optimisation

Liu, Zhesheng (2024) Singular and impulse stochastic control problems motivated by optimal harvesting and portfolio optimisation. PhD thesis, London School of Economics and Political Science.

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Abstract

In this thesis, we investigate stochastic optimal control problems motivated by (a) the optimal sustainable exploitation of an ecosystem, and (b) trading in a financial market with proportional transaction costs. In the context of optimal harvesting, we study two models. The first one is an impulse control problem with a discounted performance criterion. In this problem, the objective is to maximise a discounted performance criterion that rewards the effect of control action but involves a fixed cost at each time of a control intervention. The second problem is a singular control one, with an expected discounted criterion, an expected ergodic criterion and a pathwise ergodic criterion. We derive the explicit solutions to these stochastic control problems under general assumptions. We solve these problems by first constructing suitable solutions to their associated HJB equations. It turns out that the solution to the impulse control problem can take four qualitatively different forms, several of which have not been observed in the literature. We also show that the boundary classification of 0 may play a critical role in the solution of the problem. In the singular ergodic control problems, we develop a suitable new variational argument. Furthermore, we establish the convergence of the solution of the discounted control problem to the one of the ergodic control problems as the discounting rate function tends to zero in an Abelian sense. In the portfolio optimisation problem, we determine the growth optimal portfolio under proportional transaction costs for an investor trading a risk-free asset and a risky asset with stochastic investment opportunities given by a linear diffusion. Despite extensive research, our results are the first that construct optimal trading strategies in continuous time beyond the restrictive setting of constant parameters. This allows us to investigate the tradeoff between active trading due to the random parameters and the proportional transaction costs. We solve this problem by explicitly constructing a shadow price process and provide the asymptotic expansions of the non-trade region, the stock-cash ratio and the proportion of wealth invested in the risky asset.

Item Type:	Thesis (PhD)
Additional Information:	© 2024 Zhesheng Liu
Library of Congress subject classification:	H Social Sciences > HB Economic Theory H Social Sciences > HG Finance Q Science > QA Mathematics
Sets:	Departments > Mathematics
Supervisor:	Zervos, Mihail and Czichowsky, Christoph
URI:	http://etheses.lse.ac.uk/id/eprint/4840

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