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Title: Some examples of the spatial evolution of two-parameter processes with non-adapted initial conditions
Author: Bichard, James
ISNI:       0000 0004 2690 2790
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2009
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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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics