Research Interests
Stochastics - Uniqueness and Existence Problem for SDEs - especially McKean-Vlasov
My concentration was on
- interacting particle system
- McKean-Vlasov types of SDEs
- following research work of Professor Bossy and Professor Sznitman
Reading list: Topics in Propagation of Chaos (Sznitman), A Stochastic Particle Method for the McKean-Vlasov and the Burgers Equations (Bossy), Asymptotic Behaviour of some interacting particle systems (Sylvie) and Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process (Jourdain)
Applied Probability - Probabilistic Method for SDEs
My concentration was on
- diffusion process
- martingale problem associated with the linked PDE such as Fokker-Planck equation in McKean-Vlasov sense (losing Markov property here)
Reading list: Probability with martingales (Chris), Diffusions, Markov processes and martingales (Rogers), Some stochastic particle methods for non-linear parabolic PDEs (Bossy), Brownian Motion and Stochastic Calculus (Karatzas & Shreve)
Numerical Schemes - Monte Carlo Methods
Our focus is on
- Multilevel Monte Carlo methods applied to solve McKean-Vlasov SDEs
- Quasi-Monte Carlo
- some other variance reduction techniques and relevant machine learning algorithms
In April 2016, there was some progress in proposing a new algorithm for solving McKean-Vlasov SDEs by using MLMC rather than applying the particle system method.
Part of our result has been presented in International Conference on Monte Carlo techniques Closing conference of thematic cycle and At the Frontiers of Quantitative Finance by my supervisor Lukasz.
Reading list: Introduction to Non-parametric estimation (Tsybakov) , A Multilevel Monte Carlo Method for a class of Mckean-Vlasov processes (Ricketson) and Multilevel Monte Carlo Methods (Giles), Machine learning (Murphy)
Publications
- Ph.D. Thesis: Monte-Carlo based numerical methods for a class of non-local deterministic PDEs and several random PDE systems
- Deep learning random PDEs (preprint)
- Iterative Multi-level density estimation for McKean-Vlasov SDEs via projection joint work with Dr. Lukasz Szpruch and Prof. Denis Belomestny (submitted, Arxiv)
- Arxiv: Iterative Particle Approximation for McKean-Vlasov SDEs with Application to Multilevel Monte Carlo estimation (joint work with Dr. Lukasz Szpruch and Alvin Tse and published in Annals of Applied Probability)
- Reconstructing the Joint Probability Distribution From Basket Prices: A Mildly Ill-Posed Problem (2014, preprint)