My research interest lies in the theory and applications of probabilistic methods. Recently, I have been working on game-theoretic importance sampling (IS). Importance sampling is a variance reduction technique in Monte Carlo simulation, and can be especially effective when the quantities of interest are largely determined by rare events.
I am always interested in the theory and applications of probabilistic methods. My previous work was largely on mathematical finance, where many fundamental questions, such as option pricing and portfolio optimization, can be answered using the technical tools from stochastic optimization theory. My main contribution is the study of explicit solution to path-dependent options with a double exponential jump diffusion stock price, and the convex duality method for portfolio optimization in an incomplete semi-martingale financial market.
Recently, I have been working on game-theoretic importance sampling (IS). Importance sampling is a variance reduction technique in Monte Carlo simulation, and can be especially effective when the quantities of interest are largely determined by rare events. Its basic idea is to sample the system under a different probability distribution, and the efficiency of the algorithm depends on the choice of this alternative sampling distribution, which is the central question of importance sampling. In close collaboration with Paul Dupuis, we introduced a game theoretic approach toward importance sampling, and showed that IS algorithms are intimately connected with a differential game. The corresponding Isaacs equation and its subsolutions can serve as a very useful tool for the design and analysis of IS algorithms. The significance of this game formulation is that, for the first time, it allows one to build a comprehensive framework within which one can systematically and rigorously construct efficient dynamic IS schemes.
Much still remains to be done in the theory and applications of game-theoretic importance sampling. For example, one can consider the extension of importance sampling to different settings such as Jackson queuing networks, networks with Markov modulated arrival and service rates, and functionals of exit time and location for a small noise process in a region surrounding a stable rest point. Another interesting area I hope to investigate is the robust control for general queuing networks. There seem to be very few concrete, explicit results, especially in high dimensions. This difficulty is largely due to the dimensionality and the nature of constrained dynamics associated with such systems. I am interested in developing network models and cost structures such that a fairly complete characterization of the optimal policies are possible, regardless of dimension.
NSF grant DMS-0103669 (2001-2004). Research on stochastic optimization and applications.
NSF grant DMS-0404806 (2004-2007). Research on stochastic processes and applications.
NSF grant DMS-0706003 (2007-2010). Importance sampling and the subsolutions of an associated Isaacs Equation.
DOE grant DE-SC0002413 (2009-2011). Large deviations methods for the analysis and design of Monte Carlo schemes in physics and chemistry.
Control with partial observations and an explicit solution of Mortensen equation (with V. Benes, I. Karatzas, and D.Ocone). Appl. Math. Optim. 49 (2004), 217-239.
Option pricing under a double exponential jump diffusion models (with S. Kou). To appear in Management Sciences (2004) .
Optimal stopping with Forced exits. To appear in Math. Oper. Res. (2004).
On the convergence from discrete to continuous time in an optimal stopping problem (with P. Dupuis). To appear in Ann. Appl. Prob. (2004).
Adaptive importance sampling for uniformly recurrent Markov chains (with P. Dupuis). To appear in Ann. Appl. Prob. (2004).
A capacity expansion problem featuring exponential jump diffusion processes. Stochastics and Stochastics Reports 75 (2003), 259-274.
First passage times of a jump diffusion process (with S. Kou). Adv. Appl. Prob. 35 (2003), 504-531.
Optimal stopping with random intervention times (with P. Dupuis). Adv. Appl. Prob. 34 (2002), 1-17.
Connections between bounded variation control and Dynkin games (with I. Karatzas). Optimal Control and Partial Differential Equations 353-362 (2001). IOS Press, Amsterdam.
Utility maximization with random endowments in incomplete markets (with J. Cvitanic and W. Schachermayer). Finan. and Stoch. 5 (2001), 259-272.
Some control problems with random intervention times. Adv. Appl. Prob. 33 (2001), 402-422.
On optimal terminal wealth under transaction costs (with J. Cvitanic). J. Math. Econ. 35 (2001), 223-231.
A finite-fuel control problem with discretionary stopping (with I. Karatzas, D. Ocone, and M. Zervos". Stoch. and Stoch. Rep. 71 (2000), 1-50.
A barrier option of American type (with I. Karatzas). Appl. Math. Optim. 42 (2000), 259-280.
Discretization of deflated bond prices (with P. Glasserman). Adv. Appl. Prob. 32 (2000), 540-563.
Utility maximization with discretionary stopping (with I. Karatzas). SIAM J. on Control and Optim. 39 (2000), 306-329.
A minimization problem arising from prescribing scalar curvature functions (with L. Ma). Math. Z. 222 (1996), 1-6.
On the optimality of conditional expectation as a Bregman predictor (with A. Banerjee and X. Guo).
Importance Sampling, Large Deviations, and Differential Games (with P. Dupuis).