# David Siska

- Mean-Field / McKean Vlasov SDEs,
- Stochastic Control, including numerical algorithms,
- Mathematical Theory of Machine Learning in particular Reinforcement Learning,
- Multi-agent systems, and learning and game-theoretic aspects of those,
- Applications of the above in Financial Mathematics, DeFi, and Economics.

## 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: Twitter @dsiska, Google Scholar and MathSciNet Profile (paywalled).

## PhD Projects

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 2024) in one of my areas of interest should contact me with informal enquires: d.siska@ed.ac.uk so we can discuss a research proposal. See the School PhD applications website for available funding (we normally fully fund all students we accept) and for details regarding the formal application process. 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 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. This is joint project between Vega Protocol and 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. He is currently a Postdoc at the School of Mathematics working on Multi-agent RL in DeFi together with Simtopia.ai.
- 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. 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.

## Submitted / under review

- With A. Cartea and F. Drissi and L. Sanchez-Betancourt and L. Szpruch, Automated Market Makers Designs Beyond Constant Functions, 2023.
- With D. Sethi, The Modified MSA, a Gradient Flow and Convergence, 2022.
- 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.

## Journal Publications

- 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).
- With M. Sabate-Vidales and L. Szpruch, Unbiased deep solvers for linear parametric PDEs, Applied Mathematical Finance, 28(4), 299-329, 2021, (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, 42(3), 1991-2054, 2022, (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).

## Conference Papers (peer reviewed)

- 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).

## Talk Slides

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).

## 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.
- 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 2023/24 I will be teaching Stochastic Control and Dynamic Asset Allocation.

## Past Teaching

- I taught Stochastic Control and Dynamic Asset Allocation in 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.