Some Optimal Stopping Strategies Under Distorted Probability

In this thesis, we consider an optimal stopping problem based on a geometric Brownian motion (GBM) under the distortion of probability scale by a general non-linear function. We introduce the reformulation of the optimal stopping problem and distribution/quantile function. A GBM is given by the solution of Black-Scholes mod-el and it can be turned into an exponential martingale via a simple transformation. Thus, the Skorokhod embedding technique can be applied. The key to solving this problem consists of two parts:one is to transform the underlying asset price process to obtain a martingale; another one is to rewrite the direct expression in a distribu-tion/quantile formulation and to solve the distribution/quantile optimization problem via the Skorokhod embedding. Furthermore, we provide the proofs for those conjec-tural or unpresented claims in existing literature. In the final section, we change the constant coefficient of GBM to a more general case and use Girsanov theorem to trans-form the related underlying process to a standard GBM. Thus, we can easily solve the corresponding problem under more general financial markets.