Grinberg, Nastasiya (2011) Some problems in stochastic analysis : Itô's formula for convex functions, interacting particle systems and Dyson's Brownian motion. PhD thesis, University of Warwick.

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Abstract

This thesis consists of two main parts: Chapter 1 is concerned with studying an extension
of the Itô lemma to the convex functions. We prove that the local martingale
part of the decomposition of a convex function f of a continuous semimartingale can
be expressed in a similar way to the classical formula with the gradient of f replaced
with its subgradient. The result itself is not new, however, our approach via Brownian
perturbation is.
The second, and the largest, part of the thesis focusses on the study of a certain
family of bivariate diffusions Z(Θ,μ) = (X,R) in a wedge W = {(x, r) ∈ R x R+ : |x| ≤ rg,
parameterised by Θ ∈ (0,∞) and μ ≥ 0, with the property that X is distributed as a
Brownian motion with drift μ and R is the so-called 3-dimensional Bessel process of
drifting Brownian motion. By letting parameter Θ tend to ∞ and 0 we can recover the
two well-known couplings of the two processes coming from the Pitman’s theorem and
by considering radial part of the 3-dimensional BM (with drift μ ≥ 0) respectively. This
family of continuous processes is obtained as a diffusion approximation in Chapter 3
of a certain family of two-dimensional Markov chains arising in representation theory
and is characterised, for each Θ ∈ (0,∞) and μ ≥ 0, as a unique solution to a certain
martingale problem in Chapter 4. Moreover, we show that the process Z(μ,Θ) together
with the marginal R-process provide an example of intertwined processes. Finally, in
Chapter 5 we consider a family of certain Markov chains in the Gelfand-Cetlin cone of
depth n. We show that for n = 2 the Markov chains of Chapter 3 can be recovered. We
identify several intertwining relationships and make a step towards linking the diffusion
limit of the chain to a certain Markov function of the GUE minor process of random matrix theory, which consists of two interlaced Dyson’s Brownian motions.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Stochastic analysis, Convex functions, Brownian motion processes, Martingales (Mathematics)
Official Date:	February 2011
Dates:	Date Event February 2011 Submitted
Institution:	University of Warwick
Theses Department:	Department of Statistics
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Kendall, W. S. ; Warren, Jon
Sponsors:	University of Warwick. Dept. of Statistics
Extent:	vii, 192 leaves : ill.
Language:	eng

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