A Viscosity Solution Method for Optimal Stopping Problems with Regime Switching
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A Viscosity Solution Method for Optimal Stopping Problems with Regime Switching Yong-Chao Zhang1 · Na Zhang2
Received: 23 October 2019 / Accepted: 30 July 2020 © Springer Nature B.V. 2020
Abstract We employ the viscosity solution technique to analyze optimal stopping problems with regime switching. To be specific, we show the viscosity property of value functions with the help of dynamic programming, and more importantly, provide a mild and verifiable condition and an available bound that both can guarantee the uniqueness of viscosity solutions. Keywords Optimal stopping · Regime switching · Viscosity solution · Dynamic programming Mathematics Subject Classification (2010) 60G40 · 62L15 · 60H30
1 Introduction Regime-switching processes are appropriate candidates for describing the price of financial assets [1, 8] and the price of some commodities [7]. In addition, as Elias et al. [13] pointed out, regime-switching processes are also plausible choices of modelling the stochastic behavior of temperature. Last but not least, regime-switching processes appear in real option pricing [3]. Thus it is reasonable to consider the optimal stopping problems in which underlying processes and payoff functions are modulated by Markov chains. Many specific problems of optimal stopping with regime switching have been studied. Assuming that the stock price follows a geometric Brownian motion modulated by a twostate Markov chain, Guo [16] provided an explicit closed-form solution for Russian options. Under the same assumption as that of [16], Guo and Zhang derived an explicit closed-form
B N. Zhang
[email protected] Y.-C. Zhang [email protected]
1
School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
2
School of Mathematical Sciences, Nankai University, Tianjin 300071, China
Y.-C. Zhang, N. Zhang
solution for perpetual American options in [17] and for optimal selling rules in [18], respectively, Buffington and Elliot [5] and Yi [28] explored American options with finite maturity date, and Bensoussan et al. [2] determined optimal policies for an irreversible investment. Pemy [25] studied American options and selling rules where the price process evolves based on a regime-switching geometric Brownian motion with many states, and Liu et al. [23] and Yin et al. [29] showed some numerical methods for American options. Eloe et al. [15] developed optimal selling rules via using a regime-switching exponential Gaussian diffusion model. In a regime-switching Lévy model, Boyarchenko and Levendorskiˇi [4] showed a pricing procedure for perpetual American and real options which is efficient even though the number of states is large provided transition rates are not large with respect to riskless rates. Le and Wang [20] showed a computational procedure for a finite time horizon optimal stopping problem—it does not include the integral term and the return function is continuous, nonincreasing, convex and with a bounded support. Liu [22] priced American put option in a regime-switchin
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