On the optimal multiple stopping problem
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This report is mainly based on the paper "Optimal multiple stopping and valuation of swing options" by R. Carmona and N. Touzi (1). Here the authors model and solve optimal stopping problems with more than one exercise time. The existence of optimal stopping times is firstly proved and they then construct the value function of American put options with multiple exercises in the case of the Black-Scholes model, characterizing the exercise boundaries of the perpetual case. Finally, they extend the analysis to the swing contracts with infinitely many exercise rights. In this report, we concentrate on explaining their rigorous mathematical analysis in detail, especially for the valuation of the perpetual American put options with single exercise and two exercise rights, and the characteristics of the exercise boundaries of the multiple stopping case. These results are presented as theorems in Chapter 2 and Chapter 3.