MLMC refers to Multilevel Monte Carlo, for which I have many references available from https://people.maths.ox.ac.uk/gilesm/mlmc.html and https://people.maths.ox.ac.uk/gilesm/mlmc_community.html

These projects are all best suited to students with solid programming skills in either python or C/C++. I can assist any students who are interested in improving their C/C++ skills by using OpenMP for multi-threaded parallelisation on CPUs, or CUDA for many-threaded paralleisation on GPUs.

- Use of adjoints for the efficient estimation of Greeks, in the context of a) finite difference methods; b) Monte Carlo simulations (MCF)
- Use of MLMC branching path simulations (see arXiv paper) to reduce variance of a) bumping for Greeks, especially for digitals; b) pathwise Greeks (combining ideas from this arXiv paper); c) CDF estimation (MCF or MMSC)
- MLMC for quantile estimation following approach of Glynn and Blanchet (see arXiv paper) (MCF or MMSC)
- MLMC using approximate Lévy areas, as an alternative to the technique described in this arXiv paper (MCF or MMSC)
- MLMC change of measure for complex digital options, as suggested in my MCQMC12 review paper on multilevel Monte Carlo methods, but not yet tried. (MCF or MMSC)
- MLMC for Lévy processes, based on existing literature (MCF or MMSC)
- MLMC for randomised Numerical Linear Algebra, possibly jointly supervised by Prof Yuji Nakatsukasa (MMSC)
- A topic in High Performance Computing, using OpenMP on CPUs or CUDA on GPUs, perhaps as part of a project with another supervisor (MMSC)
- Efficient conversion of uniform random numbers to other distributions, as in this ACM TOMS paper (MMSC)