My primary research interests are in the area of probability. During June 2015, I participated in the Financial Math MRC sponsered by the AMS to help jumpstart a research program in financial math. The problems I have been working on recently are in the areas of small deviations and systemic risk, respectively.
The study of small deviations is the study of the probability that non-negative random variables take values close to zero. Small deviations results for a given process are often used to prove a Chung-like law of the iterated logarithm and a functional law of the iterated logarithm for that process. The Chung-like law of the iterated logarithm is a result about the liminf of a process as time goes to infinty. The functional law of the iterated logarithm is a result about an almost sure set of limit points for a collection of time-scaled versions of the specified process.
Currently, I am efforting to prove a Chung-like law of the iterated logarithm for Brownian motion under certain random time-change.
In current economies (both local and global), there is great interconnection between financial entities (firms) in the sense that a firm owes money (either to repay a debt or as agreed upon payments for goods or services). There is risk in such a set-up because the insolvency or default of one firm may result in the insolvency or default of other firms (since A can't pay B, B won't have the money to pay C and D).
We are interested in modelling an economy in which insolvent or defaulting firms can get loans from either other firms in the economy or a "lender of last resort" which has the funds to "bail out" any firm they wish. We seek to build a model that recognizes when sufficient capital exists to make a loan without the intervention of the "lender of last resort" and can determine the structure of such a loan, but can also recognize situations in which it is best to not grant a loan and allow the troubled firm to dissolve.
More about my research can be found in my research statement.
- Daniel Dobbs, Zachary Feinstein, Alexander Munk and Triet Pham. A dynamic model of systemic risk. In preparation
- Daniel Dobbs. A Chung-like law of the iterated logarithm for Brownian motion under certain random time-change. In preparation
- Daniel Dobbs and Tai Melcher. Small deviations for time-changed Brownian montions and applications to second-order chaos. Electronic Journal of Probability [Online]. 19(2014):1-23
- Daniel Dobbs and Tai Melcher. Smoothness of heat kernel measures on infinite-dimensional heisenberg-like Groups. Journal of Functional Analysis. 264(9), pp. 2206 - 2223