Pricing problems of perpetual Bermudan options

The pricing problem of options with an early exercise feature, such as American options, is one of the important topics in mathematical finance. The pricing formulas for American options, however, have not been found in general and the numerical methods are required to derive the price of these options, besides some exceptions, such as perpetual American options. Although the closed form pricing formula for perpetual American options in the Black and Scholes economy is known explicitly, it seems that the pricing formula for perpetual Bermudan options is not known. The value function of perpetual Bermudan options is characterized with the partial differential equation and this is solved numerically in this article