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Title: Some problems in stochastic analysis : Itô's formula for convex functions, interacting particle systems and Dyson's Brownian motion
Author: Grinberg, Nastasiya
ISNI:       0000 0004 2706 2025
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2011
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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.
Supervisor: Not available Sponsor: University of Warwick. Dept. of Statistics
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics