Our research focuses on applied probability issues in areas including financial and insurance mathematics, performance analysis and optimization, communications networks and e-commerce. One important thrust area involves development of probabilistic asymptotes and efficient Monte Carlo simulation methodologies for small but important probabilities. These include applications in finance such as measuring and controlling probability of large losses due to credit risk in a loan portfolio, due to market risk in an investment portfolio. In insurance settings we consider estimation of ruin probabilities. In networks this involves estimation of extremely small buffer overflow and large delay probabilities. In reliability systems this involves estimation of performance measures such as system unreliability and mean time to failure. Other areas of interest include developing efficient simulation methodologies to price complex financial options, simulation based ordinal optimization methodologies, computational issues in Internet auctions. From methodological point of view, we focus primarily on simulation theory, queuing theory, large deviations theory and stochastic optimization.
In the broad area of stochastic modelling and learning, the main research activities are:
(1) Stochastic control theory: This concerns dynamic optimization of performance measures associated with controlled random processes arising in engineering. Both continuous time / space and discrete time / space processes are being studied, from point of view of establishing existence of optimal control laws and characterizing the same. Of special interest are the analytically hard problems such as risk-sensitive control, average cost control, control under additional constraints, multiple time scales, stochastic dynamic games, learning algorithms for control, etc.
(2) Stochastic iterative algorithms: This broadly includes Monte Carlo type schemes and stochastic approximation algorithms. The focus is on theoretical understanding of asymptotics as well as finite time behaviour, trade-offs between variability and speed, asynchronous implementations, etc.
(3) Applications: Stochastic control and stochastic adaptive algorithms are being applied to important application areas such as power control in wireless communications, dynamic pricing, e-commerce, accelerated simulation methodologies, probabilistic methods for combinatorial optimization, etc.