Contributions to First Passage Time Problems of Brownian Motion
Let W be a standard Brownian motion with W(0) = 0 and let b be a continuous
function with b(0) > 0. The first passage time of b (from below) is then defined as the first time W crosses b (from below). A well-known method to determine the distribution of is the method of images. It gives an explicit expression for the the first hitting time distribution for all boundaries b that are the implicit solution to an integral equation which depends on some measure μ. However, the boundary b cannot be determined explicitly for most choices of μ. Therefore, the inverse method of images has emerged – at least as a numerical approach – to determine the measure μ and thus the distribution of for a given boundary b. If a measure μ exists, the boundary b will be called representable. Until now, it remained an open question for which boundaries b a measure μ exists such that b is representable. In this thesis, we present a new duality approach to the inverse method of images which enables us to give sufficient conditions for the existence of representing measures μ for concave, analytic boundaries b. Based on that, we put forward a new algorithm that improves upon existing algorithms for the inverse method of images by requiring less discretisations and giving precise approximations with very fast convergence. We also give convergence guarantees for this algorithm. As an application, we establish a connection between the existence of first passage time distributions and the representability of American options via European options. This thesis concludes by approaching the first passage time problem via Fredholm type integral equations for which we show that these uniquely determine the distribution of the first passage time.