Research |
In the past one year I have mostly worked on a family of interacting diffusions whose logarithms are Brownian motions with the following drift vector. At any time point the drift depends on the ordering (or ranking) of the values of the Brownian motions as a point process on the line. Such processes come up from various sources: econometric modeling, limits of queues, or interacting particle system modeling. The following results are the basis of two separate joint works with Sourav Chatterjee and Jim Pitman which have been submitted for publication.
For finite n, the invariant distribution of the vector of spacings between the Brownian particles can be completely described. The interest is to describe a limiting invariant distribution when n is large. We show, as n grows to infinity, a curious phenomenon occurs for the rescaled positive diffusions if we divide them by the sum of their coordinate values. Under very weak conditions, one of three things can happen to the scaled values: either they all go to zero, or the maximum grows to one while the rest go to zero, or they stabilize and converge in law to a Poisson-Dirichlet point process. The proof borrows ideas from Talagrand's analysis of Derrida's Random Energy Model of spin glasses. The results are significant in the asymptotic analysis of the Banner-Fernholz-Karatzas models of market capitalizations.
The other alternative is to start with a countable collection of diffusions. We consider one such model and discuss the similarities and differences with the previous limit. This countable model, called the infinite Atlas model, is related to the Harris model of elastic collision and the discrete Ruzmaikina-Aizenmann model for competing particles.
Prior to this, I did my PhD on convex risk measures in mathematical finance. Convex risk measures have been a recent breakthrough in the way we measure risks of financial transactions in incomplete markets. However they are difficult to deal with, and to find effective hedging strategies is a theoretical nightmare. A major work in my thesis has been to demonstarte how tools from statistical machine learning can be applied to numerically compute hedging strategies in any non-trivial situation where we cannot write down closed form expressions for the solutions. Market incompleteness or presence of arbitrage has no effect on such algorithms.
I have recently completed a project with Philip Protter which looks at option prices when the stock price under the risk neutral measure is a strict local martingale. We show that this results in several anomalies. For example, no-early-exercise is violated in many cases, raising the price of American options above the European options. The mathematics involves a study of how strict local martingales arise naturally, and how to use Doob's h-transform to effectively manipulate them.
My current project, jointly with Sourav Chatterjee, looks at particular Gaussian fields on random graphs. Taking cue from Talagrand's analysis of the diluted Sherrington-Kirkpatrick model we go on to analyze the limiting behavior of such fields. We show that the limit exists and is determined by a recursive structure if the random graph converges in the local weak convergence sense to a limiting tree.