The purpose of this thesis is to develop techniques for analysing interacting particle systems on the half-line. When the number of particles becomes large, stochastic partial differential equations (SPDEs) with Dirichlet boundary conditions will be the natural objects for describing the dynamics of the population's empirical measure. As a source of motivation, we consider systems that arise naturally as models for the pricing of portfolio credit derivatives, although similar applications are found in mathematical neuroscience, stochastic filtering and mean-field games. We will focus on a stochastic McKean--Vlasov system in which a collection of Brownian motions interact through a correlation which is a function of the proportion of particles that have been absorbed at level zero. We prove a law of large numbers where the limiting object is the unique solution to (the weak formulation of) the loss-dependent SPDE: dV<sub>t</sub>(x) = 1/2 ∂<sub>xx</sub>V<sub>t</sub>(x)dt - p(L<sub>t</sub>)∂<sub>x</sub>V<sub>t</sub>(x)dW<sub>t</sub>, V<sub>t</sub>(0)=0, where L<sub>t</sub> = 1-⎰<sup>∞</sup><sub style='position: relative; left: -.8em;'>t</sub></sup>V<sub>t</sub>(x)dx, V is a density process on the half-line and W is a Brownian motion. The correlation function is assumed to be piecewise Lipschitz, which encompasses a natural class of credit models. The first of our theoretical developments is to introduce the kernel smoothing method in the dual of the first Sobolev space, H<sup>-1</sup>, with the aim of proving uniqueness results for SPDEs. A benefit of this approach is that only first order moment estimates of solutions are required, and in the particle setting this translates into studying the particles at an individual level rather than as a correlated collection. The second idea is to extend Skorokhod's M<sub>1</sub> topology to the space of processes that take values in the tempered distributions. The benefit we gain is that monotone functions have zero modulus of continuity under this topology, so the loss process, L, is easy to control. As a final example, we consider the fluctuations in the convergence of a basic particle system with constant correlation. This gives rise to a central limit theorem, for which the limiting object is a solution to an SPDE with random transport and an additive idiosyncratic driver acting on the first derivative terms. Conditional on the systemic random variables, this driver is a space-time white noise with intensity controlled by the empirical measure of the underlying system. The SPDE has insufficient regularity for us to work in any Sobolev space higher than H<sup>-1</sup>, hence we have an example of where our extension to the kernel smoothing method is necessary.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:684934 |
| Date | January 2015 |
| Creators | Ledger, Sean |
| Contributors | Hambly, Ben |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:c46f36ee-debd-429c-80ce-d2c7e251fb83 |
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