December  2020, 25(12): 4823-4837. doi: 10.3934/dcdsb.2020128

A new weak solution to an optimal stopping problem

1. 

Center for Financial Engineering, Soochow University, Suzhou, Jiangsu 215006, China

2. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

* Corresponding author: Cong Qin

Received  October 2019 Published  April 2020

Fund Project: The first author received support from NSFC 11901416, NSF of Jiangsu BK20190812, and NSF for Universities in Jiangsu Province 19KJD100005

In this paper, we propose a new weak solution to an optimal stopping problem in finance and economics. The main advantage of this new definition is that we do not need the Dynamic Programming Principle, which is critical for both classical verification argument and modern viscosity approach. Additionally, the classical methods in differential equations, e.g. penalty method, can be used to derive some useful results.

Citation: Cong Qin, Xinfu Chen. A new weak solution to an optimal stopping problem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4823-4837. doi: 10.3934/dcdsb.2020128
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D. Bertsekas and S. Shreve, Stochastic Optimal Control: The Discrete-Time Case, Math. in Sci. and Eng., Academic Press, 1978.  Google Scholar

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A. Friedman, Variational Principles and Free-Boundary Problems, Dover Publications, 1982.  Google Scholar

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D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998.  Google Scholar

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H. Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic PDE's, Comm. Pure. Appl. Math, 42 (2009), 15-45.  doi: 10.1002/cpa.3160420103.  Google Scholar

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L. Jiang, Mathematical Modeling and Methods of Option Pricing, World Scientific Publication, 2005. doi: 10.1142/5855.  Google Scholar

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show all references

References:
[1]

D. Bertsekas and S. Shreve, Stochastic Optimal Control: The Discrete-Time Case, Math. in Sci. and Eng., Academic Press, 1978.  Google Scholar

[2]

A. Friedman, Variational Principles and Free-Boundary Problems, Dover Publications, 1982.  Google Scholar

[3]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998.  Google Scholar

[4]

H. Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic PDE's, Comm. Pure. Appl. Math, 42 (2009), 15-45.  doi: 10.1002/cpa.3160420103.  Google Scholar

[5]

L. Jiang, Mathematical Modeling and Methods of Option Pricing, World Scientific Publication, 2005. doi: 10.1142/5855.  Google Scholar

[6]

B. Oksendal, Stochastic Differential Equations, 6$^{th}$ edition, Springer, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar

[7]

H. Pham, Continuous-Time Stochastic Control and Optimization with Financial Applications, Springer, 2009. doi: 10.1007/978-3-540-89500-8.  Google Scholar

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