- Mean-Field / McKean Vlasov SDEs,
- Mathematical Theory of Machine Learning,
- Stochastic Control,
- SPDEs (Stochastic Partial Differential Equations),
- Applications in Financial Mathematics, Economics, Game Theory.
Students who studied mathematics (or applied or financial mathematics or physics or perhaps informatics with strong focus on theory) interested in working on a PhD project (to start in September 2023) in one of my areas of interest should contact me with informal enquires: email@example.com so we can discuss a research proposal. 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.
Current PhD Students
- Maria Lefter, 2018 - present, is working on McKean-Vlasov SDEs, particle approximations and stochastic control. She is funded by MIGSAA. This is part of work I do jointly with Lukasz Szpruch.
- Marc Sabate Vidales, 2019 - present, is working on machine learning algorithms for approximaiton of nonlinear and path-dependent PDEs. He is funded by The Alan Turing Institute and The Edinburgh Futures Institute. This is part of work I do jointly with Lukasz Szpruch.
- Deven Sethi, 2021 - present, is working on stochastic control. He is funded by School of Mathematics.
Former PhD Students
- Bekzhan Kerimkulov, 2017 - 2021, wrote a thesis on Iterative methods for solving stochastic optimal control problems. He was funded by MIGSAA and advised jointly also with Lukasz Szpruch. He is currently a Maxwell Institute Postoc here in Edinburgh working Theory of Reinforcement Learning.
- William Hammersley, 2016 - 2020, wrote a thesis on McKean-Vlasov SDEs with and without common noise. This was joint effort with Lukasz Szpruch. He was funded by MIGSAA. He is currently a Postdoc at Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis.
- Neelima, 2015 - 2019, wrote a thesis on nonlinear SPDEs with non-standart growth and their regularity. She was funded by University of Edinburgh's Principal's Career Development Scholarship. She is currently an Assistant Professor at Ramjas College, University of Delhi.
- With M. Lefter and L. Szpruch, Decaying derivative estimates for functions of solutions to non-autonomous SDEs, 2022.
- With M. Sabate-Vidales and L. Szpruch, Solving path dependent PDEs with LSTM networks and path signatures, 2020.
- With P. Gierjatowicz, M. Sabate-Vidales, L. Szpruch and Z. Zuric, Robust pricing and hedging via neural SDEs, 2020.
- With L. Szpruch, Gradient Flows for Regularized Stochastic Control Problems, 2020.
- With J.-F. Jabir and L. Szpruch, Mean-Field Neural ODEs via Relaxed Optimal Control, 2019.
- With B. Kerimkulov, J.-M. Leahy and L. Szpruch, Convergence of policy gradient for entropy regularized MDPs with neural network approximation in the mean-field regime, Proceedings of the 39th International Conference on Machine Learning, PMLR, 162, 12222-12252, 2022 (arXiv version).
- With M. Sabate-Vidales and L. Szpruch, Unbiased deep solvers for linear parametric PDEs, Applied Mathematical Finance, to appear, 2022 (arXiv version).
- With L. Gonon, P. Grohs, A. Jentzen and D. Kofler, Uniform error estimates for artificial neural network approximations for heat equations, IMA Journal of Numerical Analysis, to appear, 2021 (arXiv version).
- With K. Hu, Z. Ren and L. Szpruch, Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks, Ann. Inst. H. Poincaré Probab. Statist., 57(4), 2043-2065, 2021 (arXiv version).
- With W. R. P. Hammersley and L. Szpruch, McKean-Vlasov SDEs under Measure Dependent Lyapunov Conditions, Ann. Inst. H. Poincaré Probab. Statist., 57(2), 1032-1057, 2021 (arXiv version).
- With B. Kerimkulov and L. Szpruch, A modified MSA for stochastic control problems, Appl. Math. Optim., 84(3), 3417-3436, 2021 (arXiv version).
- With W. R. P. Hammersley and L. Szpruch, Weak Existence and Uniqueness for McKean-Vlasov SDEs with Common Noise, Ann. Probab., 49(2), 527-555, 2021 (arXiv version).
- With B. Kerimkulov and L. Szpruch, Exponential Convergence and stability of Howards's Policy Improvement Algorithm for Controlled Diffusions, SIAM J. Control Optim., 58(3), 1314-1340, 2020 (arXiv version).
- With Neelima, $L^p$-estimates and regularity for SPDEs with monotone semilinearity, Stoch. PDE: Anal. Comp., 8, 422-459, 2020, (arXiv version).
- With Neelima, Coercivity condition for higher order moments of nonlinear SPDEs and existence of solution under local monotonicity, Stochastics, 2019, (arXiv version).
- 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, (open access).
- 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, (open access).
- 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(10), 3719-3746, 2013.
- With E. Emmrich, Full discretization of the porous medium/fast diffusion equation based on its very weak formulation, Commun. Math. Sci., 10(4), 1055-1080, 2012.
- Error estimates for finite difference approximations of American put option price, CMAM, 12(1), 108-120, 2012, (arXiv version).
- With E. Emmrich, Full discretization of second-order nonlinear evolution equations: strong convergence and applications to elasticity theory, CMAM, 11(4),441-459, 2011.
- With I. Gyöngy, On Finite-Difference Approximations for Normalized Bellman Equations, Appl. Math. Optim., 60(3), 297-339, 2009, (arXiv version).
- With I. Gyöngy, On Randomized Stopping, Bernoulli, 14(2), 352–361, 2008, (arXiv version).
I will try to keep slides for some recent talks here.
- Convergence of Policy Gradient for Entropy Regularized MDPs with Neural Network Approximation in the Mean-Field Regime (ICML 2022).
- Neural SDEs for Robust Pricing and Hedging (CMStatistics 2021, King's College London - 18th December 2021).
- Gradient Flow for Regularized Stochastic Control Problems (LNU Stochastic Analysis Seminar - 24th November 2020).
- Learning to price and hedge path-dependent derivatives (Machine learning in finance conference, Oxford - 17th September 2019).
- Mean-field Langevin dynamics in the energy landscape of neural networks (Mittag-Leffler - May 2019, Oxford - June 2019).
- Risk-Neutral Asset Pricing (RNAP) as taught in 2016/17: Risk-Neutral Asset Pricing.
- Monte-Carlo Methods as taught in 2016/17: Monte-Carlo Methods.
- Stochastic Control and Dynamic Asset Allocation (SCDAA) as being taught in 2021/22: Stochastic Control and Dynamic Asset Allocation.
- 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
- S. Cohen, I. Gyöngy, G. dos Reis, D. Siska and L. Szpruch (eds.) Frontiers in Stochastic Analysis – BSDEs, SPDEs and their Applications, Springer, 2019.
- G. Danezis, D. Hrycyszyn, B. Mannerings, T. Rudolph, D. Siska Vega Protocol Whitepaper, 2018.
- D. Siska Incentives for Model Calibration on Decentralized Derivatives Exchanges: Consensus in Continuum, 2020.
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.
- In 2022/23 I will probably teach 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.
- In 2021/22 I taught Stochastic Control and Dynamic Asset Allocation.
- In 2020/21 I taught Stochastic Control and Dynamic Asset Allocation.
- In 2019/20 I taught Stochastic Control and Dynamic Asset Allocation.
- In 2018/19 I taught Stochastic Control and Dynamic Asset Allocation.
- 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).