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Global Closed-form Approximation of Free Boundary for Optimal Investment Stopping Problems

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  • Jingtang Ma
  • Jie Xing
  • Harry Zheng

Abstract

In this paper we study a utility maximization problem with both optimal control and optimal stopping in a finite time horizon. The value function can be characterized by a variational equation that involves a free boundary problem of a fully nonlinear partial differential equation. Using the dual control method, we derive the asymptotic properties of the dual value function and the associated dual free boundary for a class of utility functions, including power and non-HARA utilities. We construct a global closed-form approximation to the dual free boundary, which greatly reduces the computational cost. Using the duality relation, we find the approximate formulas for the optimal value function, trading strategy, and exercise boundary for the optimal investment stopping problem. Numerical examples show the approximation is robust, accurate and fast.

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  • Jingtang Ma & Jie Xing & Harry Zheng, 2018. "Global Closed-form Approximation of Free Boundary for Optimal Investment Stopping Problems," Papers 1810.09397, arXiv.org.
    • Handle: RePEc:arx:papers:1810.09397
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    References listed on IDEAS

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    1. Vicky Henderson & David Hobson, 2008. "An explicit solution for an optimal stopping/optimal control problem which models an asset sale," Papers 0806.4061, arXiv.org, revised Nov 2008.
    2. Xiongfei Jian & Xun Li & Fahuai Yi, 2014. "Optimal Investment with Stopping in Finite Horizon," Papers 1406.6940, arXiv.org.
    3. Dayanik, Savas & Karatzas, Ioannis, 2003. "On the optimal stopping problem for one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 173-212, October.
    4. Bian, Baojun & Zheng, Harry, 2015. "Turnpike property and convergence rate for an investment model with general utility functions," Journal of Economic Dynamics and Control, Elsevier, vol. 51(C), pages 28-49.
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