Structured barrier and penalty functions in infinite dimensional optimisation and analysis. Very large scale tightly-constrained optimisation problems are ubiquitous and include water management, traffic flow, and imaging at telescopes and hospitals. Massively parallel computers can solve such problems and provide physically realisable solution only if subtle design issues are mastered. Resolving such issues is the goal of this project.
Construction of near optimal oscillatory regimes in singularly perturbed control systems via solutions of Hamilton-Jacobi-Bellman inequalities. Problems of optimal control of systems evolving in multiple time scales arise in a great variety of applications (from diet to environmental modelling). This project addresses the challenge of analytically and numerically constructing rapidly oscillating controls that would 'near optimally coordinate' the slow and fast dynamics.
Perturbations in Complex Systems and Games. This project aims to: advance the perturbation theory of dynamic and stochastic games; further develop approximations of infinite dimensional linear programs by their finite dimensional counterparts, and by finding asymptotic limits of spaces of occupational measures, by solution of successive layers of fundamental equations; explain and quantify the "exceptionality" of instances of systems that are genuinely difficult to solve; and, capitalise on the ....Perturbations in Complex Systems and Games. This project aims to: advance the perturbation theory of dynamic and stochastic games; further develop approximations of infinite dimensional linear programs by their finite dimensional counterparts, and by finding asymptotic limits of spaces of occupational measures, by solution of successive layers of fundamental equations; explain and quantify the "exceptionality" of instances of systems that are genuinely difficult to solve; and, capitalise on the outstanding performance of our Snakes-and-Ladders Heuristic (SLH) for the solution of the Hamiltonian cycle problem to identify its "fixed complexity orbits" and generalise this notion to other NP-complete problems.Read moreRead less
Two-price quantitative finance. This project aims to establish a novel field, namely two-price quantitative finance, and explore its applications. The new field will integrate two major schools for modelling and explain the presence of two prices, the buying and selling prices, widely observed in the real-world markets, and the equilibrium approach from the fundamental law of one price. The outcomes would deepen our understanding of the fundamental relationship among liquidity, prices, risk and ....Two-price quantitative finance. This project aims to establish a novel field, namely two-price quantitative finance, and explore its applications. The new field will integrate two major schools for modelling and explain the presence of two prices, the buying and selling prices, widely observed in the real-world markets, and the equilibrium approach from the fundamental law of one price. The outcomes would deepen our understanding of the fundamental relationship among liquidity, prices, risk and the economy. This project expects to bring about long-term impact on quantitative finance and related applications through providing a deep understanding of, and a new perspective for, the design, risk and fairness of the finance, property and insurance markets.Read moreRead less
G-expectation and its applications to nonlinear risk management. This project will develop novel theories and methods for nonlinear risk management based on nonlinear expectations and Backward Stochastic Differential Equations. The expected outcomes of the project will place Australia in the forefront and the leading position of these fields.