The influence of geometry on stochastic processes

Abstract

We propose a class of numerical methods, called jet schemes, for solving stochastic differential equations (SDEs) whose true solutions are confined to, or attracted to, some manifold within ambient space. We give sufficient conditions for the methods to converge to the true solution as their step size tends to zero, in such a way that the numerical iterates remain near the manifold to high order. Our methods are geometrically invariant, and can be chosen to give near-perfect performance for any SDE that is diffeomorphic to n-dimensional Brownian motion. They do not require simulating iterated Itô integrals beyond those needed for the Euler–Maruyama scheme, and moreover, unlike projection-based methods, may be implemented without explicit knowledge of the solution manifold. We demonstrate the effectiveness of jet schemes in the context of the stochastic Duffing oscillator and stochastic Kepler problem. We then apply jet schemes to the Riemannian Langevin equation, an SDE which arises in the context of Monte Carlo simulation. We obtain jet-based generalisations of a Monte Carlo method known as P-MALA, taking advantage of the coordinate invariance of jet schemes. A series of numerical experiments, with both synthetic and real data, reveals that jet-based MALA is more stable than P-MALA, and also explores the state space more quickly, resulting in a higher effective sample size per sample. This resulted, for the scenarios we considered, in superior estimates of the expectation and variance of model parameters. We finish with a more theoretical chapter, in which we prove a recurrence-transience criterion for Markov chains defined on a Riemannian manifold of negative curvature. The criterion is based on certain geometric properties of the increments of the chain, defined using the Riemannian exponential map. We use our criterion to find examples of recurrent Markovian martingales on hyperbolic space of arbitrary dimension, and also on a stochastically incomplete manifold. Further, we prove that, unlike in the Euclidean case, a recurrent martingale on a manifold whose sectional curvature is bounded above by –∈ for some∈ > 0 cannot be uniformly elliptic.

Date of Award	1 Aug 2021
Original language	English
Awarding Institution	King's College London
Supervisor	John Armstrong (Supervisor)