Abstract
Abstract In this paper, we investigate dynamic optimization problems featuring both stochastic control and optimal stopping in a finite time horizon. The paper aims to develop new methodologies, which are significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature. We formulate our model to a free boundary problem of a fully nonlinear equation. Furthermore, by means of a dual transformation for the above problem, we convert the above problem to a new free boundary problem of a linear equation. Finally, we apply the theoretical results to some challenging, yet practically relevant and important, risk-sensitive problems in wealth management to obtain the properties of the optimal strategy and the right time to achieve a certain level over a finite time investment horizon. MSC:35R35, 91B28, 93E20.
Highlights
Optimal stopping problems, a variant of optimization problems allowing investors freely to stop before or at the maturity in order to maximize their profits, have been implemented in practice and given rise to investigation in academic areas such as science, engineering, economics and, finance
In the field of financial investment, an investor frequently runs into investment decisions where investors stop investing in risky assets so as to maximize their expected utilities with respect to their wealth over a finite time investment horizon
These optimal stopping problems depend on the underlying dynamic systems as well as investors’ optimization decisions
Summary
A variant of optimization problems allowing investors freely to stop before or at the maturity in order to maximize their profits, have been implemented in practice and given rise to investigation in academic areas such as science, engineering, economics and, finance. This naturally results in a mixed optimal control and stopping problem, and Ceci and Bassan [ ] is one of the typical works along this line of research. In the mathematical finance literature, choosing an optimal stopping time point is often related to a free boundary problem for a class of diffusions (see Fleming and Soner [ ] and Peskir and Shiryaev [ ]). We make a dual transformation for the problem to obtain a new free boundary problem with a linear equation.
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