Some optimal stopping problems with nontrivial boundaries for pricing exotic options (Q2774442)

The aim is to solve optimal stopping time problems in three different option cases: the financial market is supposed to be modeled with a geometric Brownian motion (Black and Scholes model): NEWLINE\[NEWLINEdX_t= X_t\mu dt+ X_t \sigma dW_tNEWLINE\]NEWLINE where \(\mu\) and \(\sigma\) are real constants and \(W\) a real Brownian motion. The first case is denoted as ``perpetual lookback'' American option, the aim is to find NEWLINE\[NEWLINE\tau\in \arg\max \bigl\{s\mapsto E[e^{-rs} (S_s-K)^+; \;s\text{ being stopping time}\bigr\},NEWLINE\]NEWLINE where \(S_t=\max_{0\leq u\leq t} X_u\), and \(r\) is a real non negative constant. The optimal solution is based on partial derivatives equations the solution of which being the value function. The other cases are to maximize in the stopping times set NEWLINE\[NEWLINE\tau\mapsto E\bigl[ e^{-rs} (l\vee X_\tau- K)^+\bigr]NEWLINE\]NEWLINE and finally NEWLINE\[NEWLINE\tau \mapsto E[e^{-rs}l \vee X_\tau]NEWLINE\]NEWLINE where \(l\) and \(r\) are positive real constants \((l>0)\). Explicit value functions are done in the three cases, depending on the solution to a differential equation, respectively depending on constants which are a solution to implicit equations.