Bichard, James (2009) Some examples of the spatial evolution of two-parameter processes with non-adapted initial conditions. PhD thesis, University of Warwick.

PDF WRAP_THESIS_Bichard_2009.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (842Kb)

Abstract

The central result of this thesis is an enlargement of filtrations result for the filtration (Fx; x ≥ 0), where Fx = σ{Bys : y ≤ x, s ∈ [0,∞)} and (Bxt; x ∈ R, t ∈ [0,∞)) is a Brownian sheet on a complete probability space. Although this is a fairly straightforward extension of a result presented in [Yor97] for Brownian filtrations, it is of use to us in a couple of applications. The first is a discussion of ‘bridged’ Brownian sheets, in which we try to describe the law of a Brownian sheet which is fixed along some curve in the parameter space. The second application is a study of the spatial evolution of solutions to the stochastic heat equation. We fix a starting point in space, and describe the spatial evolution as driven by an (Fx; x ≥ 0)-adapted noise. Unfortunately, we find that the initial condition is not in F0. If we add this initial information to (Fx; x ≥ 0), the driving noise is no longer a martingale, but our enlargement result allows us to write a semimartingale decomposition, in some sense. We are in fact able to write a system of stochastic differential equations which describe the spatial evolution of solutions, such that each equation is driven by a martingale with respect to this larger filtration.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Brownian bridges (Mathematics) -- Research, Stochastic differential equations -- Numerical solutions, Martingales (Mathematics)
Date:	June 2009
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Assing, Sigurd, 1965-
Format of File:	pdf
Extent:	155 leaves : charts
Language:	eng
URI:	http://wrap.warwick.ac.uk/id/eprint/3119

Request changes to a record

Actions (login required)

View Item