Abstract:
We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. Apart from their immediate application to the computation of distribution of first exit times, these transformations turn out to be instrumental in resolving the long-standing issue of non-uniqueness for the Black-Scholes equations in derivative pricing. Using an appropriate diffusion transformation we show that the Black-Scholes price is the unique solution of an alternative Cauchy problem. Finally, we use these path transformations to propose a unified framework for solving explicitly the optimal stopping problem for one-dimensional diffusions, which in particular is relevant for the pricing and optimal exercise boundaries of perpetual American options.