I am a Reader (Associate Professor) at the School of Mathematics, University of Edinburgh.

I am Scientific Director of the Finance and Economics Programme at the Alan Turing Institute, London where I am also a Research Fellow. Before moving to Edinburgh, I was a Nomura Junior Research Fellow at the Institute of Mathematics, University of Oxford, and a member of the Oxford-Man Institute for Quantitative Finance. I hold a Ph.D. in Mathematics and Statisitcs from the University of Strathclyde in Glasgow.I am currently researching on: mathematics of deep learning, mean-field models, Game Theory and decision making under uncertainty, sampling and optimisation algorithms, optimal transport and theory of gradient flows.

My broad research interests focus on probability theory, stochasitc analysis and theoretical machine leanrning.

At the Turing Insitute, I co-organise the interest groups: Machine Learning in Finance and Sampling algorithms for data analytics

For updates on my research see my Google Scholar page. You may also check my profile on the Research Gate or the MathSciNet

- Kerimkulov B., Siska D., Szpruch L., A modified MSA for stochastic control problems,
*arXiv* - Siska D., Szpruch L., Gradient Flows for Regularized Stochastic Control Problems,
*arXiv* - Majka M.B., Sabate-Vidales M., Szpruch L., Multi-index Antithetic Stochastic Gradient Algorithm,
*arXiv* - Ni H., Szpruch L., Wiese M., Liao S., Xiao B., Conditional Sig-Wasserstein GANs for Time Series Generation,
*arXiv* - Arribas I.P., Salvi C., Szpruch L., Sig-SDEs model for quantitative finance,
*arXiv* - Jabir JF., Siska D., Szpruch L., Mean-Field Neural ODEs via Relaxed Optimal Control,
*arXiv* - Duncan A., Nusken N., Szpruch L., On the geometry of Stein variational gradient descent,
*arXiv* - Belomestny D., Szpruch L., Shuren T., Iterative Multilevel density estimation for McKean-Vlasov SDEs via projections,
*arXiv* - Kaitong H., Zhenjie R., Siska., Szpruch L., Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks,
*arXiv* - Szpruch L., Tse A., Antithetic multilevel particle system sampling method for McKean-Vlasov SDEs,
*arXiv* - Chassagneux JF., Szpruch L., Tse A., Weak quantitative propagation of chaos via differential calculus on the space of measures,
*arXiv* - Sabate M., Siska D, Szpruch L., Unbiased deep solvers for parametric PDEs,
*arXiv* - Lionnet A., dos Reis G. and Szpruch L., Full-Projection scheme for monotone BSDEs with polynomial growth,
*arXiv* - Hammersley R. P. W, Siska D., Szpruch L., McKean-Vlasov SDEs under Measure Dependent Lyapunov Conditions,
*arXiv* - Nagapetyan T, Duncan A.B., Hasenclever L., Vollmer S.J., Szpruch L, Zygalakis L, The True Cost of Stochastic Gradient Langevin Dynamics
*arXiv*

- Hammersley R. P. W, Siska D., Szpruch L., Weak Existence and Uniqueness for McKean-Vlasov SDEs with Common Noise, to appear in Annals of Probability
*arXiv* - Kerimkulov B., Siska D., Szpruch L., Exponential Convergence and stability of Howards's Policy Improvement Algorithm for Controlled Diffusions, SIAM J. Control Optim., 58(3), 1314–1340. journal,
*arXiv* - Majka M.B., Mijatovic A., Szpruch L, Non-asymptotic bounds for sampling algorithms without log-concavity, to appear in Annals of Applied Probability,
*arXiv* - Giles M.B., B. Majka M.B., Szpruch L, Vollmer S., Zygalakis K., Multilevel Monte Carlo methods for the approximation of invariant measures of stochastic differential equations, Statistics and Computing, 2019
*arXiv* - Szpruch L., Tan S., Tse A., Iterative Particle Approximation for McKean-Vlasov SDEs with application to Multilevel Monte Carlo estimation, Annals of Applied Probability
Volume 29, Number 4 (2019), 2230-2265. journal,
*arXiv* - Neuenkirch A., Szolgyenyi M, Szpruch L., An adaptive Euler-Maruyama scheme for stochastic differential equations with discontinuous drift and its convergence analysis, SIAM J. Numer. Anal. 57-1 (2019), pp. 378-403. journal,
*arXiv* - AlRachid H., Bossy M., Ricci C., Szpruch L., New particle representations for ergodic McKean-Vlasov SDEs, ESAIM: ProcS
Volume 65, 2019 journal,
*arXiv* - Szpruch, L. Zhang, X. V-Integrability, Asymptotic Stability And Comparison Theorem of Explicit Numerical Schemes for SDEs,
Mathematics of Computations 87 (2018), 755-783 ,
*arXiv* - Lionnet A., dos Reis G. and Szpruch L., Convergence and qualitative properties of modified explicit schemes for BSDEs with polynomial growth
, Annals of Applied Probability. Volume 28, Number 4 (2018), 2544-2591.,
*journal**arXiv* - Lionnet, A., dos Reis, G and Szpruch, L., Time discretisation of FBSDE with polynomial growth drivers and reaction-diffusion PDEs, Annals of Applied Probability, Vol. 25, 2563-2625, Number 5, 2015.
*arXiv* - Giles, M.B. and Szpruch, L., Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L\'{e}vy area simulation, Annals of Applied Probability, Vol. 24, 1585-1620 , Number 4 2014.
*arXiv* - Neuenkirch, A. and Szpruch, L., First order strong approximations of scalar SDEs with values in a domain, Numerische Mathematik, Vol. 128-1, pp 103-136, 2014.
*arXiv* - Higham, D.J., Mao, X. and Szpruch, L. Convergence, non-negativity and stability of a new Milstein Scheme with applications to finance, DCDS-B,18(8):2083 - 2100, AIMS, 2013
*arXiv* - Giles, M.B. and Szpruch, L, Antithetic multilevel Monte Carlo estimation of financial options in Monte Carlo and Quasi-Monte Carlo Methods 2012.
*arXiv* - Cohen, S.N. and Szpruch, L. On Markovian Solutions to Markov Chain BSDEs Numerical Algebra, Control and Optimization, 2012, 2(2):257-269
*arXiv* - Cohen, S.N. and Szpruch, L. A limit order book model for latency arbitrage, Mathematics and Financial Economics, 6(3):211-227, 2012.
*arXiv* - Dereich, S., Neuenkirch, A. and Szpruch, L., An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process, Proceedings of the Royal Society A, Vol. 468, No. 2140, pp. 1105–1115, 2012.
*arXiv* - Mao X., and Szpruch L., Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, Journal of Computational and Applied Mathematics, 238:14-28,2013
*arXiv* - Szpruch, L. and Mao. X., Strong convergence rates for backward Euler-Maruyama method for dissipative-type stochastic differential equations with super-linear diffusion coefficients, Stochastics, 85, no. 1, 144171, 2013.
*preprint* - Higham, D. J. , Intep, S., Mao, X., and Szpruch, L., Hybrid Simulation of auto-regulation within transcription and translation, BIT Numer Math., Vol. 51, No. 1, pp. 177-196, 2011.
- Szpruch, L. Mao X., Higham, D. J., and Pan, J., Numerical simulation of a strongly nonlinear Ait- Sahalia-type interest rate model, BIT Numer Math, Vol. 51, No. 2, pp. 405-425, 2011.
- Wu, F., Mao, X. and Szpruch L., Almost sure exponential stability of numerical solutions for stochastic delay differential equations, Numerische Mathematik, Vol. 115, No. 4, pp. 681-697, 2010.
- Szpruch, L., Higham, D. J., Comparing hitting time behavior of Markov jump processes and their diffusion approximations, Multiscale Model. Simul., No. 8, pp. 605-621, 2010.

- Giles, M.B. and Szpruch, L, Multilevel Monte Carlo methods for applications in finance, in: Gerstner, Kloeden (Eds.), Recent Advances in Computational Finance, World Scientific, 2013.
*arXiv*

- Jean-Francois Jabir Research assosiate (2019-present)
- Patryk Gierjatowicz, - PhD student (2018-present) funded by MIGSAA , jointly with David Siska
- Bekzhan Kerimkulov, - PhD student (2017-present) funded by MIGSAA , jointly with David Siska
- Shuren Tan - PhD student (2015-present)
- William Hammersley - PhD student (2017-present) funded by MIGSAA , jointly with David Siska

- Nikolas Nüsken ( (2018) Postdoc at Turing jointly with Andrew Duncan
- Alvin Tse - PhD student (2016-2019)
Thesis "Quantitative propagation of chaos of McKean-Vlasov equations via the master equation"

- Xiling Zhang - PhD student (2013-2017), jointly with Sandy Davie

Lukasz Szpruch

The University of Edinburgh

James Clerk Maxwell Building

Peter Guthrie Tait Road

Edinburgh EH9 3FD

E-mail: l.szpruch[at]ed.ac.uk