My research interests are in financial mathematics and applied
probability, especially in the context of optimal stochastic control.
My two primary areas of work concern
There are numerous applications of the above models; some that I have worked on in detail include:
- Optimal switching problems. These are repeated optimal stopping
models that can also be viewed as a simplified impulse control setting.
They have wide applications in resource management, decision making
under uncertainty and exotic derivatives.
- Stochastic control under partial observations. Many operations
research and financial applications feature agents that have limited
knowledge of the stochastic environment. Thus, the problems they face
require simultaneous estimation and control. I develop new computational
algorithms to make this possible in a robust but tractable framework.
Financial and Insurance Mathematics
- Valuation of exotic energy contracts. I am particularly
interested in complex contracts such as power tolling agreements,
natural gas storage. Recently, I have also started looking at
cap-and-trade emissions trading; these models require combination of
game theory, stochastic control and classical option pricing.
- Stochastic non-zero sum games. I am interested in stochastic differential games where the agents are non-cooperative but do not directly hurt each other. The structure and solution of such games is still poorly understood. I investigate the structure of feedback Nash equilibria in e.g. non-zero-sum stopping games.
- Liquidity modeling. The standard Black and Scholes model does
not take into account limited market liquidity. I am interested in
quantifying this limitation, especially through the indifference pricing
- Optimal risk sharing with distortion risk measures. Insurance
markets bring together participants to exchange their risks. Without any
specified risk exchange mechanism a priori, it is observed empirically
that deductible insurance (or tranching) emerges as the
optimal contract choice. We attempt to formally prove such phenomena by
deriving optimal risk sharing contracts within the class of distortion
(or spectral) risk preferences.
- Stochastic mortality models. There is an ongoing move away
from classical actuarial models of deterministic hazard rates towards
the more realistic stochastic hazard rate framework. This includes
borrowing tools from credit and fixed income derivatives and applying
them in the insurance context.
Sequential Estimation and Control Models
- Management of Disease Outbreaks. The aim of public health policies is to design cost-efficient strategies to mitigate impact of disease outbreaks. Policy-makers must make early decisions in the face of significant uncertainty about severity and other outbreak parameters. I construct a dynamic control model that combines online estimation with sequential decision making within a Monte Carlo framework. An important case study is analysis of seasonal influenza outbreaks/pandemics on a regional level.
- Robust disorder problems. The Bayesian formulation of a change-point problem involves monitoring the observation process to detect a change in the underlying signal. Multiple objectives regarding balancing the observation/delay costs and early detection are assumed. Understanding the resulting control problem is important to build further sequential detectors.
- Tracking problems. A signal process X follows piecewise deterministic dynamics (ie is a jump-Markov process). The aim of the controller is to track as closely as possible X using several noisy/incomplete observation channels. This generic problem is a stepping stone to multiple models of interest, including Hidden Markov Model formulations of disease outbreaks, tracking of multiple security violations and regime-switching models in finance.
- Monte Carlo methods for Snell envelopes is the main computational engine for all of the above models. A range of groundbreaking tools have been developed in the past 15 years, originally motivated by American options problems. However, given the more complex problems presented above and new computational capabilities (such as parallel computing, novel regression methods, advanced importance sampling tools), a new generation of models is in the making.
PStat Department Seminar
Activity Group on Financial Mathematics
Upcoming Meetings where we can meet:
Fourth Western Conference in Mathematical Finance (WCMF), Los Angeles, CA
June 6-7, 2011
Third International Workshop on Sequential Methodologies (IWSM), Palo Alto, CA
June 14-16, 2011
Seventh International Congress on Industrial and Applied Mathematics (ICIAM), Vancouver, BC
July 18-22, 2011
46th Actuarial Research Conference (ARC), Storrs, CT
August 11-13, 2011
Research in Options (RiO) Workshop, Angra Dos Reis, Brazil
Nov 28-Dec 1, 2010
SIAM FM10 Conference on Financial Mathematics and Engineering, San Francisco, CA
November 19-20, 2010.
Sixth World Congress of the Bachelier Society, Toronto, ON.
June 22-26, 2010.
14th International Congress on Insurance: Mathematics and Economics, Toronto, ON
Workshop on Computational Methods in Finance, Toronto, ON
March 22-24, 2010.
IPAM New Directions in Financial Mathematics, Los Angeles, CA
January 5-9, 2010
Third Western Conference on Mathematical Finance, Santa Barbara, CA
INFORMS Applied Probability Conference, Ithaca, NY.
July 12-15, 2009
Optimal Stopping and Applications Symposium, Turku, Finland.
June 23-26, 2009
MSRI Workshop on Economic Games and Mechanisms to Address Climate Change, Berkeley, CA.
May 4-6, 2009
Fifth Princeton-Oxford Workshop on Financial Mathematics, Princeton, NJ
March 27-28, 2009
14th UCEI Power Conference, Berkeley, CA.
March 20, 2009.
Second SIAM Conference on Financial Mathematics & Engineering, FM08, New Brunswick, NJ
November 21-22, 2008
Western Conference on Mathematical Finance, WCMF08, Austin, TX
October 31-November 2, 2008
Workshop on Optimization and Optimal Control, Linz, Austria
October 20-24, 2008
International Symposium on Business and Industrial Statistics, ISBIS2008, Prague, Czech Republic
July 2-July 4, 2008
13th International Symposium on Dynamic Games, ISDG08, Wroclaw, Poland
June 30-July 2, 2008
13th International Symposium on Dynamic Games, ISDG08, Wroclaw, Poland
June 30-July 3, 2008
NSF/CBMS Conference on Convex Duality Methods, UCSB, CA
June 26-June 30, 2008
Stochastic Processes and Applications Conference, SPA07, Urbana, IL
August 6-10, 2007
Workshop on Mathematics and the Environment: Energy Risk, Environmental Uncertainty and Public Decision Making, Banff, Canada
May 8-12, 2007
Workshop on Financial Engineering for Actuarial Mathematics, Ann Arbor, MI
May 4-6, 2007
INFORMS 2006 Annual Meeting, Pittsburgh, PA.
Nov 5-8, 2006.
Bachelier Finance Society Fourth World Congress, Tokyo, Japan.
August 17-20, 2006.
SIAM Conference on Financial Mathematics and Engineering FM06, Boston,
July 9-12, 2006.
Workshop on Optimization Problems in Financial Economics, Banff, Canada.
Risk Management Conference: Integrated Risk Management in Operations
and Global Supply Chain Management, Ann Arbor, MI.
June 2-4, 2006.
Çinlar Day and Seminar on Stochastic Processes, Princeton, NJ.
Symposium on Optimal Stopping with Applications, Manchester, UK.