Reflected diffusions and piecewise diffusion approximations of Levy processes

Abstract

In the first part of the thesis, the solvability of stochastic differential equations with reflecting boundary conditions is studied. Such equations arise in singular stochastic control problems as a way for determining the optimal strategies. The stochastic differential equations represent homogeneous one-dimensional diffusions while the boundaries are given by c`adl`ag functions. Pathwise solutions are constructed under mild assumptions on the coefficients of the equations. In particular, the solutions are derived as the diffusions’ scale functions composed with appropriately time-changed reflected Brownian motions. Several probabilistic properties are addressed and analysed. In the second part of the thesis, piecewise diffusion approximations of Levy processes are studied. Such approximating processes have been called Itˆo semi-diffusions. While keeping the statistical fit to Levy processes, this class of processes has the analytical tractability of Ito diffusions. At a given time grid, their distribution is the same as the one of the underlying Levy processes. At times outside the grid, they evolve like homogeneous diffusions. The analysis identifies conditions under which Itˆo semi-diffusions can be used as an alternative to Levy processes for modelling financial asset prices. In particular, for a sequence of Itˆo semi-diffusions determined by a given Levy process, conditions for the convergence of their finite-dimensional distributions to the ones of the Levy process are established. Furthermore, for a single Ito semi-diffusion, conditions for the existence of pricing measures are established.