On spatially irregular ordinary differential equations and a pathwise volatility modelling framework

LaTeX source of Ryan McCrickerd's PhD thesis (Department of Mathematics, Imperial College London, 2021), and a Lean 4 formalisation of its main results.

• arXiv: 1902.01673 (PDF)
• Imperial College London thesis repository: doi:10.25560/92202
• Author: Ryan McCrickerd, ORCID 0009-0006-8496-7594

Abstract

This thesis develops a new framework for modelling price processes in finance, such as an equity price or foreign exchange rate. This can be related to the conventional Itô calculus-based framework through the time integral of a price's squared volatility, or 'cumulative variance'. In the new framework, corresponding processes are strictly increasing, solve random ordinary differential equations (ODEs), and are composed with geometric Brownian motion. The new framework has no dependence on stochastic calculus, so processes can be studied on a pathwise basis using probability-free ODE techniques and functional analysis.

The ODEs considered depend on continuous driving functions which are 'spatially irregular', meaning they need not have any spatial regularity properties such as Hölder continuity. They are however strictly increasing in time, thus temporally asymmetric. When sensible initial values are chosen, initial value problem (IVP) solutions are also strictly increasing, and the solution set of such IVPs is shown to contain all differentiable bijections on the non-negative reals. This enables the modelling of any non-negative volatility path which is not zero over intervals, via the time derivative of solutions. Despite this generality, new well-posedness results establish the uniqueness of solutions going forwards in time. A condition is provided which prohibits explosions, and then the IVPs' solution map is shown to be continuous with respect to uniform convergence over compacts.

Motivation to explore this framework comes from its connection with a time-changed Heston volatility model. The framework shows how Heston price processes can converge to a generalisation of the normal-inverse Gaussian (NIG) Lévy process, and reveals a deeper relationship between integrated Cox-Ingersoll-Ross (CIR) processes and the inverse Gaussian (IG) process. Within this framework, a 'Riemann-Liouville-Heston' (RLH) martingale model is defined which generalises these relationships to fractional counterparts. This model's implied volatilities are simulated, and exhibit features characteristic of leading volatility models.

Citing

Please cite the thesis. GitHub's "Cite this repository" button draws on CITATION.cff.

@phdthesis{McCrickerd_2021,
  author        = {McCrickerd, Ryan},
  title         = {On spatially irregular ordinary differential equations and a pathwise volatility modelling framework},
  school        = {Imperial College London},
  year          = {2021},
  doi           = {10.25560/92202},
  eprint        = {1902.01673},
  archiveprefix = {arXiv},
}

Building

Requires a recent TeX Live with latexmk; the source was last built with TeX Live 2025.

latexmk        # writes main.pdf
latexmk -c     # remove intermediate files

latexmk reruns BibTeX, regenerating main.bbl from bib/thesis.bib, whenever the bibliography changes. main.bbl is tracked because arXiv compiles from it.

Lean formalisation

lean/ formalises the thesis's main results in Lean 4 with Mathlib, with the results they rest on. lean/README.md lists them, and lean/NOTES.md records where the Lean proofs differ from the thesis's.

Layout

main.tex            document skeleton
main.bbl            bibliography as generated by BibTeX
tex/
  preamble.tex      packages, page layout and macros
  frontmatter.tex   title page, abstract, declarations, acknowledgements, contents
  prologue.tex
  1_introduction.tex
  2_wellposed.tex
  3_solutions.tex
  4_framework.tex
  5_conclusion.tex
  epilogue.tex
  notation.tex
  appendix.tex      RLH simulation code
bib/
  thesis.bib        bibliography database
  thesis.bst        bibliography style, a modified copy of apa-good.bst
figures/            figures, as PDF
lean/               Lean formalisation
CITATION.cff        citation metadata

Licence

The thesis is licensed under CC BY-NC 4.0, as stated in its copyright declaration. See LICENSE. The Lean formalisation in lean/ is licensed under the Apache License 2.0; see lean/LICENSE.