New Approaches to Modelling and Analysing Long-Memory Random Processes. The project aims to develop new approaches using infinite-dimensional Markov processes to solving important long-standing problems from the theory of long memory random processes and their applications. It aims to: construct a class of new representations of random processes; derive inequalities for the key numerical characteristics; and, devise and investigate numerical methods for computing the characteristics and for perf ....New Approaches to Modelling and Analysing Long-Memory Random Processes. The project aims to develop new approaches using infinite-dimensional Markov processes to solving important long-standing problems from the theory of long memory random processes and their applications. It aims to: construct a class of new representations of random processes; derive inequalities for the key numerical characteristics; and, devise and investigate numerical methods for computing the characteristics and for performing statistical inference on the long memory models. The accuracy of numerical approximations will be analysed using simulations on supercomputers. Expected outcomes include models and results of practical importance with applications such as intrusion detection problems, cell motility for biological data and telecommunication.Read moreRead less
Mathematical studies on the statistical properties of complex systems. Introduced in the late `50's to model nuclear spectra, random matrices are now standard in the theory of quantum chaos, mesoscopic phenomena and disordered systems. These are all examples of physical complex systems, characterized by unknown interactions leading to predictable behaviour due to symmetries. Vast mathematical structures result from the symmetries - integrable systems, Painleve equations, Macdonald polynomial the ....Mathematical studies on the statistical properties of complex systems. Introduced in the late `50's to model nuclear spectra, random matrices are now standard in the theory of quantum chaos, mesoscopic phenomena and disordered systems. These are all examples of physical complex systems, characterized by unknown interactions leading to predictable behaviour due to symmetries. Vast mathematical structures result from the symmetries - integrable systems, Painleve equations, Macdonald polynomial theory to name a few. These structures will be further developed, leading to the analytic form of distribution functions quantifying classes of complex systems. Analogous statistical quantification is the essence of recently proposed methods to analyze artificial complex systems such as the stock market.Read moreRead less
Discovery Early Career Researcher Award - Grant ID: DE210101352
Funder
Australian Research Council
Funding Amount
$330,000.00
Summary
Inverting the Signature Transform for Rough Paths and Random Processes. The signature transform provides an effective summary of the essential information encoded in multidimensional paths that are highly oscillatory and involve complicated randomness. The main goal of this project is to develop new algorithmic methods to reconstruct rough paths and random processes from the signature transform at various quantitative levels. This project expects to make theoretical breakthrough on the significa ....Inverting the Signature Transform for Rough Paths and Random Processes. The signature transform provides an effective summary of the essential information encoded in multidimensional paths that are highly oscillatory and involve complicated randomness. The main goal of this project is to develop new algorithmic methods to reconstruct rough paths and random processes from the signature transform at various quantitative levels. This project expects to make theoretical breakthrough on the significant open problem of signature inversion, thereby advancing knowledge in the areas of rough path theory and stochastic analysis. The newly developed methods will be utilised in combination with the emerging signature-based approach to study important problems in financial data analysis and visual speech recognition.
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