- Mean-Field / McKean Vlasov SDEs,
- Mathematical Theory of Machine Learning,
- Stochastic Control,
- SPDEs (Stochastic Partial Differential Equations),
- Applications in Financial Mathematics, Economics, Game Theory.
School of Mathematics, University of Edinburgh
Room 4611, JCMB, King's Buildings
Tel. 0131 651 9091
Students interested in working on a PhD project (to start in September 2020) in one of my areas of interest should contact me with informal enquires: email@example.com. See the School PhD applications website and MAC-MIGS website for available funding (we normally fully fund all students we accept) and for details regarding the formal application process (feel free to contact me to discuss research plan before submitting the application).
Current PhD Students
- Neelima, 2015 - present, she is working on nonlinear SPDEs with non-standart growth and their regularity. She is funded by University of Edinburgh's Principal's Career Development Scholarship.
- Bekzhan Kerimkulov, 2017 - present, he is working on PDEs on measure spaces, McKean-Vlasov SDEs and stochastic control. He is funded by MIGSAA. This is part of work I do jointly with Lukasz Szpruch.
Here are some events I am (or was) helping to organize.
- ICMS Workshop on Wasserstein Calculus and Related Topics: 19th - 23rd November 2018.
- MIGSAA Mini-Course: Singular SPDEs and Regularity Structures: 26th - 30th June 2018.
- International Workshop on BSDEs, SPDEs and their Applications: 3rd-7th July 2017.
- Conference on Stochastic Analysis in Honor of Istvan Gyöngy's 65th Birthday: 10th-12th September 2016.
- Maxwell Institute Probability Day: 13th May 2016.
- With K. Hu, Z. Ren and L. Szpruch, Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks (2019).
- With B. Kerimkulov and L. Szpruch, Exponential Convergence and stability of Howards's Policy Improvement Algorithm for Controlled Diffusions (2018).
- With M. Sabate-Vidales and L. Szpruch, Martingale Functional Control Variates via Deep Learning (2018).
- With W. Hammersley and L. Szpruch, McKean-Vlasov SDEs under Measure Dependent Lyapunov Conditions (2018).
- With Neelima, $L^p$-estimates and regularity for SPDEs with monotone semilinearity (2017).
- With Neelima, Coercivity condition for higher order moments of nonlinear SPDEs and existence of solution under local monotonicity, Submitted (2016).
- With I. Gyöngy, Itô Formula for Processes Taking Values in Intersection of Finitely Many Banach Spaces, Stoch. PDE: Anal. Comp., 5(3), 428-455, 2017.
- With E. Emmrich, Nonlinear stochastic evolution equations of second order with damping, Stoch. PDE: Anal. Comp., 5(1), 81-112, 2017. (arXiv version).
- With I. Gyöngy and S. Sabanis, Convergence of tamed Euler schemes for a class of stochastic evolution equations, Stoch. PDE: Anal. Comp., 4(2), 225-245, 2016.
- With E. Emmrich and A. Wroblewska-Kaminska, Equations of second order in time with quasilinear damping: existence in Orlicz spaces via convergence of a full discretisation, Math. Methods Appl. Sci., 39(10), 2449-2460, 2016. (preprint version).
- With E. Emmrich and M. Thalhammer, On a full discretisation for nonlinear second-order evolution equations with monotone damping: construction, convergence, and error estimates, Found. Comput. Math. 2015
- With E. Emmrich, Evolution equations of second order with nonconvex potential and linear damping: existence via convergence of a full discretization, J. Diff. Eq., 255 (2013), no. 10, 3719-3746
- With E. Emmrich, Full discretization of the porous medium/fast diffusion equation based on its very weak formulation , Commun. Math. Sci., 10 (2012), no. 4, 1055-1080
- Error estimates for finite difference approximations of American put option price, CMAM, 12 (2012), no. 1, 108-120
- With E. Emmrich, Full discretization of second-order nonlinear evolution equations: strong convergence and applications to elasticity theory, CMAM, 11 (2011), no. 4, 441-459
- With I. Gyöngy, On Finite-Difference Approximations for Normalized Bellman Equations, Appl. Math. Optim., 60 (2009), no. 3, 297-339 (arXiv version)
- With I. Gyöngy, On Randomized Stopping, Bernoulli, 14 (2008), no. 2, 352–361
- PhD. Thesis: Numerical approximations of stochastic optimal stopping and control problems
- MSc. Thesis: Stochastic Differential Equations Driven by Fractional Brownian Motion – a White Noise Distribution Theory Approach
- Lecture notes for RNAP as taught in 2016/17: Risk-Neutral Asset Pricing.
- Lecture notes for MCM as taught in 2016/17: Monte-Carlo Methods.
- Lecture notes for SCDAA as taught in 2018/19: Stochastic Control and Dynamic Asset Allocation.
- In 2018/19 I am teaching Stochastic Control and Dynamic Asset Allocation.
- I am supervising student projects for the MSc in Computational and Financial Mathematics and MSc in Financial Modelling and Optimization.
Numerical Computation Output Examples
- The solution of a stochastic Ginzburg-Landau equation in one spatial dimension using a newly developed Tamed Euler scheme (see related article)
- The payoff of an American put option, obtained using finite difference method, with the optimal exercise boundary indicated and a sample path of the price process (see related article)
- In 2017/18 I taught Stochastic Control and Dynamic Asset Allocation.
- In 2017/18 I was one of the team teaching OOPA (materials on Learn).
- In 2017/18 I was one of the team teaching SMSTC Foundations of Probability for PhD students.
- In 2016/17 I taught Monte Carlo Methods and Simulation.
- In 2016/17 taught Risk Neutral Asset Pricing.
- In 2016/17 taught Object-Oriented Programming with Applications .
- In 2016/17 I was one of the team teaching SMSTC Probability I for PhD students.
- In 2015/16 I taught Risk Neutral Asset Pricing.
- In 2015/16 I taught Object-Oriented Programming with Applications .
- In 2015/16 I was one of the team teaching SMSTC Probability I for PhD students.
- In 2014/15 I taught Risk Neutral Asset Pricing (requires UoE login).