The design and use of stochastic methods, or Monte Carlo methods, in computational finance is what interests me most lately. A good example of sophisticated Monte Carlo algorithms in finance are those designed for pricing American options. Increasing the efficiency of Monte Carlo algorithms, by the use of variance reduction techniques and low-discrepancy sequences, is an ongoing interest of mine. See my paper with Warren Eastman, "Randomized quasi-Monte Carlo methods in pricing securities", Journal of Economic Dynamics and Control, for a discussion of some of these issues.
Monte Carlo and quasi-Monte Carlo methods
I worked on problems regarding variation reduction techniques ("Error Reduction Techniques in Quasi-Monte Carlo Integration", Mathematical and Computer Modelling), error estimation in quasi-Monte Carlo methods ("Random Sampling from Low-Discrepancy Sequences: Applications to Option Pricing",
Mathematical and Computer Modelling, and
"Error Estimation for Quasi-Monte Carlo Methods", Monte Carlo and Quasi-Monte Carlo Methods 1996, Lecture Notes in Statistics, Springer-Verlag), and the design of hybrid methods that combine stochastic techniques from Monte Carlo and deterministic techniques from quasi-Monte Carlo, in some elaborate way ("A Probabilistic Result on the Discrepancy of a Hybrid-Monte Carlo Sequence and Applications", Monte Carlo Methods and Applications). Another application area I worked on is solving systems of linear equations using Monte Carlo methods ("Solving Linear Equations by Monte Carlo Methods" SIAM Journal on Scientific Computing).
For more on my research, visit my personal web page and check out my publications!