Abstract
The author studies the optimal investment stopping problem in both continuous and discrete cases, where the investor needs to choose the optimal trading strategy and optimal stopping time concurrently to maximize the expected utility of terminal wealth. Based on the work of Hu et al. (2018) with an additional stochastic payoff function, the author characterizes the value function for the continuous problem via the theory of quadratic reflected backward stochastic differential equations (BSDEs for short) with unbounded terminal condition. In regard to the discrete problem, she gets the discretization form composed of piecewise quadratic BSDEs recursively under Markovian framework and the assumption of bounded obstacle, and provides some useful a priori estimates about the solutions with the help of an auxiliary forward-backward SDE system and Malliavin calculus. Finally, she obtains the uniform convergence and relevant rate from discretely to continuously quadratic reflected BSDE, which arise from corresponding optimal investment stopping problem through above characterization.
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Acknowledgements
The author would like to thank her Ph.D. supervisors, Prof. Shanjian Tang and Dr. Gechun Liang for their guidance and help during the research and revision of this paper, and also the anonymous referees and the editor for their careful reading and helpful comments.
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This work was supported by the China Scholarship Council, the National Science Foundation of China (No. 11631004) and the Science and Technology Commission of Shanghai Municipality (No. 14XD1400400).
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Sun, D. The Convergence Rate from Discrete to Continuous Optimal Investment Stopping Problem. Chin. Ann. Math. Ser. B 42, 259–280 (2021). https://doi.org/10.1007/s11401-021-0256-7
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DOI: https://doi.org/10.1007/s11401-021-0256-7
Keywords
- Optimal investment stopping problem
- Utility maximization
- Quadratic reflected BSDE
- Discretely reflected BSDE
- Convergence rate