David Siska

I am a Reader in the School of Mathematics with research interests in:

• Mathematics of reinforcement learning,
• Stochastic control, especially numerical methods,
• Stochastic analysis, PDEs, McKean-Vlasov SDEs,
• Applications of the above in engineering, ecoonomics and finance / insurance.

Contact Details

School of Mathematics, University of Edinburgh
Room 4611, JCMB, King's Buildings
Tel. 0131 651 9091
Email. d.siska@ed.ac.uk
Online presence: @dsiska.bsky.social, Google Scholar and MathSciNet Profile (paywalled).

Prospective PhD students interested in working with me should read some of my recent papers and the information provided here before emailing me.

I apologise but I don't have capacity to supervise any summer projects (except for UoE students and even in that case there's no guarantee).

Current PhD Students

• Deven Sethi, 2021 - present, is working on stochastic control. He is funded by the School of Mathematics.
• Lukasz Sliwinski, 2021 - present, is working learning algorithms for multi-agent systems and their convergence. He is funded by the MAC-MIGS doctoral training centre.
• Galen Cao, 2022 - present, is working multi-agent modelling for limit-order-book driven derivates markets. Funding is currently from the MAC-MIGS doctoral training centre.

Former PhD Students

• Marc Sabate Vidales, 2019 - 2023, wrote a thesis on Machine learning methods in Mathematical Finance. He was funded by The Alan Turing Institute and The Edinburgh Futures Institute.
• Maria Lefter, 2018 - 2022, wrote a thesis on McKean-Vlasov SDEs and the long time behaviour of derivatives of associated PDE solutions and uniform-in-time particle approximations of McKean-Vlasov SDEs. She was funded by MIGSAA.
• 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.
• 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.
• 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.

New / Under Review

• With R.-A. Lascu and L. Szpruch, PPO in the Fisher-Rao geometry, 2025. You can also see the video of FR-PPO learning the car racing gym environment.
• With D. Sethi, Mirror descent for constrained stochastic control problems, 2025.
• With J. Cao, Ergodic optimal liquidations in DeFi, 2024.
• With R.-A. Lascu and M. B. Majka and L. Szpruch, Linear convergence of proximal descent schemes on the Wasserstein space, 2024.
• With J. Cao and L. Szpruch and T. Treetanthiploet, Logarithmic regret in the ergodic Avellaneda-Stoikov market making model, 2024.
• With A. Cartea and F. Drissi and L. Sanchez-Betancourt and L. Szpruch, Automated Market Makers Designs Beyond Constant Functions, 2023.

Peer-Reviewed Publications (journals and conferences)

1. With B. Kerimkulov and L. Szpruch and Y. Zhang, Mirror Descent for Stochastic Control Problems with Measure-valued Controls, Stoch. Process. their Appl., 2025, (arXiv version).
2. With D. Sethi and Y. Zhang, Entropy annealing for policy mirror descent in continuous time and space, SIAM J. Control Optim., 63(4), 3006-3041, 2025, (arXiv version).
3. With B. Kerimkulov and J.-M. Leahy and L. Szpruch and Y. Zhang, A Fisher-Rao gradient flow for entropy-regularised Markov decision processes in Polish spaces, Found. Comput. Math., 2025, (arXiv version).
4. With D. Sethi, The Modified MSA, a Gradient Flow and Convergence, Ann. Appl. Probab., 34(5), 4455-4492, 2024, (arXiv version).
5. With L. Szpruch, Gradient Flows for Regularized Stochastic Control Problems, SIAM J. Control Optim., 62(4), 2036-2070, 2024, (arXiv version).
6. With P. Gierjatowicz, M. Sabate-Vidales, L. Szpruch and Z. Zuric, Robust pricing and hedging via neural SDEs, Journal of Computational Finance, 26(3), 1755-2850, 2023, (arXiv version).
7. 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).
8. With C. Constantinescu, A. Dias, B. Li, and S. Wang, Effect of Stop-Loss Reinsurance on Primary Insurer Solvency, Risks, 10(10), 2022, (preprint version).
9. With M. Sabate-Vidales and L. Szpruch, Unbiased deep solvers for linear parametric PDEs, Applied Mathematical Finance, 28(4), 299-329, 2021, (arXiv version).
10. 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, 42(3), 1991-2054, 2022, (arXiv version).
11. 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).
12. 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).
13. With B. Kerimkulov and L. Szpruch, A modified MSA for stochastic control problems, Appl. Math. Optim., 84(3), 3417-3436, 2021, (arXiv version).
14. 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).
15. 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).
16. With Neelima, $L^p$-estimates and regularity for SPDEs with monotone semilinearity, Stoch. PDE: Anal. Comp., 8, 422-459, 2020, (arXiv version).
17. With Neelima, Coercivity condition for higher order moments of nonlinear SPDEs and existence of solution under local monotonicity, Stochastics, 2019, (arXiv version).
18. 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).
19. With E. Emmrich, Nonlinear stochastic evolution equations of second order with damping, Stoch. PDE: Anal. Comp., 5(1), 81-112, 2017, (arXiv version).
20. 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).
21. 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)..
22. 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.
23. 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.
24. 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.
25. Error estimates for finite difference approximations of American put option price, CMAM, 12(1), 108-120, 2012, (arXiv version).
26. With E. Emmrich, Full discretization of second-order nonlinear evolution equations: strong convergence and applications to elasticity theory, CMAM, 11(4),441-459, 2011.
27. With I. Gyöngy, On Finite-Difference Approximations for Normalized Bellman Equations, Appl. Math. Optim., 60(3), 297-339, 2009, (arXiv version).
28. With I. Gyöngy, On Randomized Stopping, Bernoulli, 14(2), 352–361, 2008, (arXiv version).

Talk Slides

I will try to keep slides for some recent talks here.

• Mirror Descent for MDPs, controlled diffusions and entropy annealing (Banff workshop on Advances in Stochastic Control and Reinforcement Learning 2025). There is a video recording of my talk but even better there are recordings of most of the other talks.
• Automated Market Makers Designs beyond Constant Functions (Finance and Stochastics Seminar, Imperial College London 2023).
• Fisher-Rao Gradient Flow for Entropy Regularised MDPs in Polish Spaces (Seminaire DFF-FiME, Institut Henri Poincare 2023).
• 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).

Lecture Notes

• 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.

Other

• 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 J.-F. Jabir and L. Szpruch, Mean-Field Neural ODEs via Relaxed Optimal Control, 2019.
• D. Siska Incentives for Model Calibration on Decentralized Derivatives Exchanges: Consensus in Continuum, 2020.
• G. Danezis, D. Hrycyszyn, B. Mannerings, T. Rudolph, D. Siska Vega Protocol Whitepaper, 2018.
• S. Cohen, I. Gyöngy, G. dos Reis, D. Siska and L. Szpruch (eds.) Frontiers in Stochastic Analysis – BSDEs, SPDEs and their Applications, Springer, 2019.
• 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

Teaching

• In 2025/26 I will be teaching Research Skills in Financial Mathematics (RSFM) in Semester 2.

Past Teaching

• I taught Stochastic Control and Dynamic Asset Allocation in 2024/25, 2023/24, 2022/23, 2021/22, 2020/21, 2019/20, 2018/19 and in 2017/18.
• I taught Risk-Neutral Asset Pricing in 2016/17, 2015/16 and in 2014/15.
• In 2016/17 I taught Monte Carlo Methods and Simulation.
• I taught Object Oriented Programming with Applications in 2017/18, 2016/17 and in 2015/16.
• I was one of the team teaching the graduate course SMSTC Foundations of Probability in 2017/18, 2016/17 and in 2015/16.