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.
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.
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.
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.
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.
This is a numerical verification that the Multilevel Milstein treatment also works well for basket options.
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.
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.
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.
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.
This paper continues the collaboration with Rob Scheichl and Aretha Teckentrup at the University of Bath.
This is a survey article looking at the application of multilevel methods in computational finance.
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.
This is a survey article for the proceedings of MCQMC12 based on my plenary lecture.
This paper is an extension to paper #13, including an approximation to Lévy areas to efficiently compute digital and barrier options.
This paper extends MLMC analysis to the estimation of cumulative distribution functions and probability densities.
This is a 70-page review article -- MATLAB code for all of the test cases presented is available here.
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.
This short paper explains that it is easy to use adaptive timestepping within multilevel Monte Carlo, for both SDEs and continuous-time Markov processes.
This paper is a continuation of the previous collaboration.
These 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.
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.
This paper analyses the MLMC variance for financial options based on exponential Lévy models.
This paper looks at MLMC for particle dispersion modelling. In particular it develops a more efficient MLMC treatment of particle reflections at a boundary.
This 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.
This paper summarises the theoretical results in our earlier papers.
This 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.
This 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.
This paper discusses progress and future research possibilities in applying MLMC ideas to nested expectations.
This is a theoretical paper which uses a computational cost model proportional to the number of random bits which are used.
This 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.
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.
This 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.
This is a second theoretical paper which uses a computational cost model proportional to the number of random bits which are used.
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).
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.
This paper also looks at the problem of approximating invariant measures of SDEs.
This 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.
This 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.
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.
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 develops a wavelet representation of white noise to provide a consistent Multilevel Monte Carlo coupling for improved variance reduction.
Chaotic SDEs pose a challenge to sensitivity analysis because sensitivities grow exponentially. In this paper we avoid this problem by building on prior research in which we introduced a novel coupling between coarse and fine paths in a multilevel Monte Carlo formulation.
In this paper we combine a standard MLMC treatment of SDEs with the use of approximate Normal random variables which can be generated extremely cheaply. The paper includes an analysis of the mixed 4-way difference with different timesteps and true or approximate Normal random variables. Even bigger computational savings may be achieved in other settings where it is extremely costly to invert a CDF to generate random variables from uniform random variables.
This paper extends our previous work on financial risk estimation by developing a MLMC method with a cost which is insensitive to the number of products in a financial portfolio.
This paper is a follow-on to #45 with a concentration on different approaches to approximating the inverse cumulative Normal distribution, and their suitability and implementation on different hardware. It also has numerical experiments using an approximation to the inverse of the non-central chi-squared distribution for simulating the CIR SDE.
This article develops a new approach for improving the multilevel variance for discontinuous functionals of the terminal state of an SDE solution. Inspired by talks on branching diffusions, a deterministic repeated branching is used to greatly reduce the variance at the expense of a very modest increase in sample cost.
This article, in which I was a very minor contributor, expands on #36 and #37 by considering triply-nested expectations!
This review article, following a plenary presentation at MCQMC2020, discusses the many methods in the literature to improve the MLMC variance convergence which is usually poor when the output functional has a discontinuity.