Stochastic processes on curved spaces

Preview

PDF WRAP_Theses_Mramor_2021.pdf - Submitted Version - Requires a PDF viewer. Download (1MB) | Preview

Request Changes to record.

Abstract

We investigate semimartingales and other classes of stochastic processes on smooth manifolds. First, we introduce continuous semimartingales on smooth manifolds and review a well-established theory surrounding it, including the notion of Brownian motion on Riemannian manifolds. Second, we focus on the Brownian motion on a unit sphere of arbitrary dimension and consider the projected process consisting of several of the components of the original spherical Brownian motion. We obtain a stochastic differential equation the projection satisfies and find its invariant measure. We study a more general stochastic differential equation on the unit ball and show the existence and path-wise uniqueness of the solution and we show that the solution enjoys a skew-product decomposition analogous to the classical skew-product decomposition of Euclidean Brownian motion. Third, a variant of the skew-product decomposition exists for the full spherical Brownian motion and features Wright-Fisher diffusion process as the radial process. Since Wright-Fisher diffusion processes can be simulated exactly as in [JS17], we utilise the rapid-spinning phenomenon of the skew-product decomposition and obtain a novel algorithm for the exact simulation of the increments of spherical Brownian motion. We analyse the algorithm numerically and obtain the range of parameters where it remains stable and show that it then outperforms previous simulation algorithms from the literature. Fourth, we define a Lévy process on a smooth manifold with a connection as a projection of a solution of a Marcus stochastic differential equation on the holonomy bundle, driven by the holonomy-invariant Euclidean Lévy process. We show that Lévy processes on manifolds are Markov processes and characterize them via their infinitesimal generators. In order to prove this characterization we obtain the path-wise construction of the stochastic horizontal lift and stochastic anti-development of a general discontinuous semimartingale, which generalizes the result in [PE92] to manifolds with non-unique geodesics between distinct points. We compare our definition with previous definitions of Lévy processes on Lie groups and Riemannian manifolds and provide several examples.

Request changes or add full text files to a record

Repository staff actions (login required)

View Item