Optimal stopping problems for maxima and minima in models with asymmetric information

We derive closed-form solutions to optimal stopping problems related the pricing of perpetual American withdrawable standard and lookback put call options in an extension Black-Merton-Scholes model with asymmetric information. It is assumed that contracts are withdrawn by their writers at last hitting times for underlying risky asset price its running maximum or minimum over infinite time interval which not respect observable filtration. show exercise first process reaches some lower upper stochastic boundaries depending on current values minimum. The proof based reduction original necessarily two-dimensional associated free-boundary means smooth-fit normal-reflection conditions. prove maximal minimal first-order nonlinear ordinary differential equations.