Kai Liu works in stochastic analysis. Specific topics of interest include:
- Stability of infinite dimensional stochastic systems - such topics as stochastic control, population biology and turbulence can give rise to such systems. Stability is often regarded as the first characteristic to study.
- Large deviations of stochastic systems - the study of probabilities of very rare events, which may be interpreted as catastrophies.
Relevant mathematical techniques include stochastic calculus; large deviation theory; calculus (ODE, PDE); functional analysis; measure theory.
- The convex analytic approach to controlled Markov jump processes and Markov decision processes - this approach, dual to the dynamic programming approach, is convenient for the study of constrained problems.
- Optimal policies in multicriteria problems.
- Communication networks – in particular, Active Queue Management models of TCP connection. With a varying input stream of jobs, given that one can control the input intensity, the goal is to find an optimal control policy.
- Controlled Markov processes with local transitions - such processes appear in Queuing Theory, Reliability, Epidemiology, etc. In some cases, the underlying model can be approximated by a diffusion, or a deterministic ordinary differential equation. It is then of interest to evaluate the accuracy of such approximations and to elaborate optimal control policies.
Relevant mathematical techniques include probability and random processes; calculus (ODE, PDE); functional analysis; optimisation; computer programming (Matlab, Maple).
Kamila Żychaluk works on non-parametric statistics, Bayesian statistics, and applications in biology. Specific topics of interest include:
- Kernel, wavelet and local polynomial estimation - traditional statistical techniques estimate an unknown function by assuming some parametric form, eg Linear Regression, Quadratic Regression, Logistic Regression. Nonparametric methods aim to fit a quite general (smooth) function to the observed data.
- Applications in biology - eg coral reef development, psychometric response functions.
Relevant mathematical techniques include nonparametric statistics; generalized linear models; bootstrap; Bayesian statistics; statistical computing (R, Matlab). Software for model-free estimation of a psychometric function may be found at http://www.modelfree.liv.ac.uk/index.html