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UBC Theses and Dissertations

Monotone optimal policies for quasivariational inequalities arising in optimal portfolio liquidation Crawford, Daniel J.


This thesis studies the Hamilton-Jacobi-Ballman quasivariational inequality (HJBQVI), the corresponding optimal value function, and discrete schemes useful for approximating this value function. Moreover, the structural properties of the optimal policy of particular discrete scheme is studied. The motivation is to find a convergent, approximating scheme for the otherwise complicated HJBQVI that has monotone policy structure that can be exploited in a stochastic gradient estimation scheme to approximate optimal policy function parameters. In order to motivate this approach, we consider the problem of optimal liquidation of a single risky asset portfolio as an impulse control problem. The model is defined over continuous time, state, and compact action sets, and the optimal liquidation value and strategy are found from the viscosity solution of a HJBQVI. It is shown that the optimal strategy is monotone in the number of shares owned and the time remaining to liquidation. This structural result is exploited to estimate the optimal policy via a reinforcement learning method based on the simultaneous perturbation stochastic approximation (SPSA) algorithm. The optimal policy can be estimated without knowledge of the parameters of the underlying model.

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