My main research on Monte Carlo methods concerns the development of multilevel methods.

Inspired by multigrid ideas for the iterative solution of discretised PDEs, this can be viewed as a recursive control variate approach which combines stochastic simulations with differing levels of resolution. This results in a significant reduction in the order of complexity, the computational cost required to obtain a user-specified accuracy.

- M.B. Giles.
'Multi-level Monte Carlo path simulation'.
*Operations Research*, 56(3):607-617, 2008. (PDF)This is my original multilevel paper. It presents numerical results for SDEs using an Euler discretisation, but also analyses the computational complexity for a general class of methods and applications, assuming that the discretisation satisfies certain conditions.

The MATLAB code used to produce the figures for the paper is available here.

- M.B. Giles.
`Improved multilevel Monte Carlo convergence using the Milstein scheme'.
343-358, in
*Monte Carlo and Quasi-Monte Carlo Methods 2006*, Springer, 2008. (PDF)This paper presents significantly improved numerical results using the Milstein discretisation. The Milstein method's improved strong convergence leads to most of the computational effort being confined to the coarsest levels.

The MATLAB code used to produce the figures for the paper is available here.

- M.B. Giles, D.J. Higham and X. Mao.
'Analysing multilevel Monte Carlo for options with non-globally
Lipschitz payoff'.
*Finance and Stochastics*, 13(3):403-413, 2009. (PDF)This was a collaboration with Des Higham and Xuerong Mao from the University of Strathclyde, in which we performed a numerical analysis of the multilevel Euler-Maruyama method in the first paper.

- M.B. Giles and B.J. Waterhouse.
'Multilevel quasi-Monte Carlo path simulation'. pp.165-181 in
*Advanced Financial Modelling*, in*Radon Series on Computational and Applied Mathematics*, de Gruyter, 2009. (PDF)This was a collaboration with Ben Waterhouse of the University of New South Wales. It uses randomised quasi-Monte Carlo techniques based on a rank-1 lattice rule to further improve the computational efficiency.

- M.B. Giles.
`Multilevel Monte Carlo for Basket Options'.
Winter Simulation Conference '09.
(PDF)
This is a numerical verification that the Multilevel Milstein treatment also works well for basket options.

- K.A. Cliffe, M.B. Giles, R. Scheichl, A.L. Teckentrup,
'Multilevel Monte Carlo Methods and Applications to Elliptic PDEs
with Random Coefficients',
*Computing and Visualization in Science*, 14(1):3-15, 2011. (PDF)This is a collaboration with Rob Scheichl and Aretha Teckentrup at the University of Bath, and Andrew Cliffe at the University of Nottingham. This applies the multilevel approach to elliptic SPDEs which arise in the modelling of nuclear waste repositories, with the permeability of the rock being modelled as a log-Normal stochastic field.

- Y. Xia, M.B. Giles.
`Multilevel path simulation for jump-diffusion SDEs', pp.695-708 in
*Monte Carlo and Quasi-Monte Carlo Methods 2010*, Springer, 2012. (PDF)This paper with my student Yuan Xia tackles Merton-style jump-diffusion models. The key feature of this paper is the use of a change of measure to cope with cases in which the jump rate is path-dependent which would otherwise lead to jumps at different times on coarse and fine paths.

- S. Burgos, M.B. Giles.
`Computing Greeks using multilevel path simulation', pp.281-296 in
*Monte Carlo and Quasi-Monte Carlo Methods 2010*, Springer, 2012. (PDF)This paper with my student Sylvestre Burgos deals with the calculation of sensitivities. This involves differentiating the payoff, and the loss of smoothness causes difficulties for the multilevel method.

- M.B. Giles, C. Reisinger.
'Stochastic finite differences and
multilevel Monte Carlo for a class of SPDEs in finance',
*SIAM Journal of Financial Mathematics*, 3(1):572-592, 2012. (PDF)This is a collaboration with my colleague Christoph Reisinger. It is another SPDE application, but in this case it is an unusual parabolic SPDE which arises in a financial credit modelling application. One key aspect of this paper is the proof of mean square stability.

- A.L. Teckentrup, R. Scheichl, M.B. Giles, E. Ullmann.
Further analysis of multilevel Monte Carlo
methods for elliptic PDEs with random coefficients',
*Numerische Mathematik*, 125(3):569-600, 2013. (PDF)This paper continues the collaboration with Rob Scheichl and Aretha Teckentrup at the University of Bath.

- M.B. Giles, L. Szpruch.
'Multilevel Monte Carlo methods for applications in finance', in
*Recent Developments in Computational Finance*, World Scientific, 2013. (PDF)This is a survey article looking at the application of multilevel methods in computational finance.

- M.B. Giles, L. Szpruch.
'Antithetic multilevel Monte Carlo estimation for multi-dimensional
SDEs without Lévy area simulation',
*Annals of Applied Probability*, 24(4):1585-1620, 2014. (PDF)This paper addresses the use of the Milstein approximation in multiple dimensions. This usually requires the simulation of Lévy areas, but we have developed an antithetic technique which gives a high rate of multilevel convergence without simulating Lévy areas, and this paper includes a lengthy numerical analysis of this.

- M.B. Giles.
'Multilevel Monte Carlo methods', pp.79-98 in
*Monte Carlo and Quasi-Monte Carlo Methods 2012*, Springer, 2014. (PDF)This is a survey article for the proceedings of MCQMC12 based on my plenary lecture.

- M.B. Giles, L. Szpruch.
'Antithetic multilevel Monte Carlo estimation for multidimensional SDEs',
pp.297-312 in
*Monte Carlo and Quasi-Monte Carlo Methods 2012*, Springer, 2014. (PDF)This paper is an extension to paper #13, including an approximation to Lévy areas to efficiently compute digital and barrier options.

- C. Lester, C. Yates, M.B. Giles, R.E. Baker.
'An adaptive multi-level simulation algorithm for stochastic biological systems'.
*Journal of Chemical Physics*, 142(2):2015. (PDF) - M.B. Giles, T. Nagapetyan, K. Ritter.
'Multilevel Monte Carlo approximation of distribution functions and densities'.
*SIAM/ASA Journal on Uncertainty Quantification*, 3:267-295, 2015. (PDF).This paper extends MLMC analysis to the estimation of cumulative distribution functions and probability densities.

- M.B. Giles.
'Multilevel Monte Carlo methods'.
*Acta Numerica*, 24:259-328, 2015. (PDF)This is a 70-page review article -- MATLAB code for all of the test cases presented is available here.

- F. Vidal-Codina, N.C. Nguyen, M.B. Giles, J. Peraire.
'A model and variance reduction method for computing statistical
outputs of stochastic elliptic partial differential equations'.
*Journal of Computational Physics*, 297:700-720, 2015. (PDF)This paper is slightly unusual in using MLMC in an application where there is not a natural geometric sequence of levels. Instead, it determines empirically the best sequence of levels to use.

- M.B. Giles, C. Lester, J. Whittle.
'Non-nested adaptive timesteps in multilevel Monte Carlo computations'.
*Monte Carlo and Quasi-Monte Carlo Methods 2014*, Springer, 2015. (PDF)This short paper explains that it is easy to use adaptive timestepping within multilevel Monte Carlo, for both SDEs and continuous-time Markov processes.

- F. Vidal-Codina, N.C. Nguyen, M.B. Giles, J. Peraire.
'An empirical interpolation and model-variance reduction method
for computing statistical outputs of parametrized stochastic
partial differential equations'.
*SIAM/ASA Journal on Uncertainty Quantification*, 4(1):244-265, 2016. linkThis paper is a continuation of the previous collaboration.

- C. Lester, R.E. Baker, M.B. Giles, C.A. Yates.
'Extending the multi-level method for the simulation of stochastic
biological systems'.
*Bulletin of Mathematical Biology*, 78(8):1640-1677, 2016. link - W. Fang, M.B. Giles.
'Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz
drift: Part I, finite time interval'.
arXiv pre-print, 2016.
link

W. Fang, M.B. Giles. 'Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift: Part II, infinite time interval'. arXiv pre-print, 2017. linkThese two papers analyse adaptive time-stepping for SDEs with a drift which is not globally Lipschitz. This follows on from the earlier paper with Lester and Whittle.

- M.B. Giles, T. Nagapetyan, K. Ritter.
'Adaptive multilevel Monte Carlo approximation of distribution functions'.
arXiv pre-print, 2017.
link
This paper extends our previous paper on this topic by developing a fully-automated adaptive procedure to select the key parameters in the MLMC algorithm.

- M.B. Giles, Y. Xia.
'Multilevel Monte Carlo for exponential Lévy models'.
*Finance and Stochastics*, 21(4):995-1026, 2017. linkThis paper analyses the MLMC variance for financial options based on exponential Lévy models.

- G. Katsiolides, E.H. Muller, R. Scheichl, T. Shardlow, M.B. Giles,
D.J. Thomson.
'Multilevel Monte Carlo and improved timestepping methods in
atmospheric dispersion modelling'.
*Journal of Computational Physics*, 354:320-343, 2018. linkThis paper looks at MLMC for particle dispersion modelling. In particular it develops a more efficient MLMC treatment of particle reflections at a boundary.

- M.B. Giles.
'MLMC for Nested Expectations'.
*Festschrift for 80th Birthday of Ian Sloan*, Springer, 2018. linkThis paper discusses the application and analysis of MLMC methods for different kinds of nested expectations.

- M.B. Giles, F.Y. Kuo, I.H. Sloan.
'Combining sparse grids, multilevel MC and QMC for elliptic PDEs
with random coefficients'.
*Monte Carlo and Quasi-Monte Carlo Methods 2016*, Springer, 2018. linkThis paper has a number of meta-theorems (similar to the original MLMC theorem) which look at the complexity of various MLMC/MLQMC generalisations. It also contains a number of ideas of variants of the Multi-Index Monte Carlo (MIMC) method.

- W. Fang, M.B. Giles.
'Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift'.
*Monte Carlo and Quasi-Monte Carlo Methods 2016*, Springer, 2018. linkThis paper summarises the theoretical results in our earlier papers.

- M.B. Giles, F. Bernal.
'Multilevel estimation of expected exit times and other functionals
of stopped diffusions'.
*SIAM/ASA Journal on Uncertainty Quantification*, 6(4):1454-1474, 2018. linkThis paper considers SDEs in a bounded space-time domain, and associated path functionals with expected values which are equivalent to parabolic PDE solutions through the Feynman-Kac theory.

- M. Croci, M.B. Giles, M.E. Rognes, P.E. Farrell.
'Efficient white noise sampling and coupling for multilevel
Monte Carlo with non-nested meshes'.
*SIAM/ASA Journal on Uncertainty Quantification*, 6(4):1630-1655, 2018. linkThis paper develops and analyses a technique for the efficient sampling of white noise realizations in order to generate Gaussian fields with a Matern covariance structure. This includes coupled constructions for coarse and fine grids for use within a multilevel Monte Carlo simulation.

- M.B. Giles.
'MLMC for nested expectations'.
pp.425-442 in
*Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan*, Springer, 2018. linkThis paper discusses progress and future research possibilities in applying MLMC ideas to nested expectations.

- M.B. Giles, M. Hefter, L. Mayer, K. Ritter.
'Random bit quadrature and approximation of distributions on
Hilbert spaces'.
*Foundations of Computational Mathematics*, 19(1):205-238, 2019. linkThis is a theoretical paper which uses a computational cost model proportional to the number of random bits which are used.

- M.B. Giles, T. Goda.
'Decision-making under uncertainty: using MLMC for efficient
estimation of EVPPI'.
*Statistics and Computing*, 29(4):739-751, 2019. linkThis paper develops and analyses an MLMC approach to the estimation of Expected Value of Partial Perfect Information, which is relevant to applications such as the evaluating the cost-effectiveness of medical research.

- M.B. Giles, K. Debrabant, A. Roessler.
'Numerical analysis of multilevel Monte Carlo path
simulation using the Milstein discretisation'.
*Discrete and Continuous Dynamical Systems - series B*, 24(8):3881-3903, 2019. link - W. Fang, M.B. Giles.
'Multilevel Monte Carlo method for ergodic SDEs without contractivity'.
*Journal of Mathematical Analysis and Applications*, 476(1):149-176, 2019. linkThis paper couples coarse and fine paths using a "spring" to improve the MLMC variance in cases in which the two paths would otherwise diverge exponentially.

- M.B. Giles, M. Hefter, L. Mayer, K. Ritter.
'Random bit multilevel algorithms for stochastic differential equations'.
*Journal of Complexity*, 54:101395, 2019. linkThis is a second theoretical paper which uses a computational cost model proportional to the number of random bits which are used.

- M.B. Giles, A.L. Haji-Ali.
'Multilevel nested simulation for efficient risk estimation'.
*SIAM/ASA Journal on Uncertainty Quantification*, 7(2):497-525, 2019. link.This paper develops and analyses an adaptive multilevel Monte Carlo algorithm for nested simulation problems arising the calculation of VaR (Value-at-Risk) and CVaR (Conditional Value-at-Risk, also known as Expected Shortfall).

- M.B. Giles, A.L. Haji-Ali.
'Sub-sampling and other considerations for efficient risk estimation
in large portfolios'. arXiv pre-print, 2019,
link.
This paper follows on from the previous paper by tackling the computational cost of very large portfolios by sub-sampling within them. It also looks at the use of various control variates which reduce the MLMC variance.

- M.B. Giles, M.B. Majka, L. Szpruch, S.J. Vollmer, K.C. Zygalakis.
'Multi-level Monte Carlo methods for the approximation of
invariant measures of stochastic differential equations'.
*Statistics and Computing*, 30(3):507-524, 2020. link.This paper also looks at the problem of approximating invariant measures of SDEs.

- M.B. Giles, M. Hefter, L. Mayer, K. Ritter.
'An Adaptive Random Bit Multilevel Algorithm for SDEs',
in
*Multivariate Algorithms and Information-Based Complexity*, De Gruyter, 2020. linkThis is the third paper using a computational cost model proportional to the number of random bits which are used. In this case, the numerical algorithm uses a Brownian Bridge construction with more precision for the terms with the greatest span. The numerical results indicate a near-optimal order of complexity.

- W. Fang, M.B. Giles.
'Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift'.
*Annals of Applied Probability*, 30(2):526-560, 2020. linkThis is the published version of papers 22a/b, proving that the use of adaptive timesteps cures the instability of the standard uniform timestep Euler-Maruyama method when applied to SDEs with a drift which is not globally Lipschitz.

- T. Hironaka, M.B. Giles, T. Goda, H. Thom.
'Multilevel Monte Carlo estimation of the expected value of sample information'.
*SIAM/ASA Journal on Uncertainty Quantification*, 8(3):1236-1259, 2020. link,This paper extends the previous work on the estimation of Expected Value of Partial Perfect Information, relevant to applications such as the evaluating the cost-effectiveness of medical research.

- M.B. Giles, O. Sheridan-Methven.
'Analysis of nested multilevel Monte Carlo using approximate Normal
random variables'. arXiv pre-print, 2021.
link,
This paper defines a nested MLMC algorithm for SDE path simulations based on Brownian path increments which are only approximately Normal, using various approximations to the inverse of the Normal CDF. The paper includes a detailed analysis of the variance of the resulting MLMC estimators.

- W. Fang, Z. Wang, M.B. Giles, C.H. Jackson, N.J. Welton, C. Andrieu,
H. Thom. 'Multilevel and Quasi Monte Carlo methods for the
calculation of the Expected Value of Partial Perfect Information'.
*Medical Decision Making*, to appear in 2021. link,This paper continues the research on the estimation of EVPPI, looking at the use of Quasi-Monte Carlo methods as well as MLMC.

This paper performs a numerical analysis of the multilevel Milstein method presented in paper #2, and also presents basket option results similar to those in paper #5.

Current MLMC research, involving several collaborations, is addressing the following applications:

- stochastic PDEs
- nested simulations
- use of approximate probability distributions

- EP/E031455/1: Development of Multilevel Monte Carlo Algorithms for Mathematical Finance
- EP/H05183X/1: Multilevel Monte Carlo Methods for Elliptic Problems with Applications to Radioactive Waste Disposal
- MA/3630057: Inference, COmputation and Numerics for Insights into Cities (ICONIC)