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Title: Discrete approximations in stochastic rough path theory
Author: Flint, Guy Henry
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2016
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We consider two discrete schemes for studying and approximating stochastic differential equations (SDEs) using the theory of rough paths. The first part of the thesis introduces a canonical method for transforming a discrete sequential data set into an associated rough path made up of lead-lag increments. In particular, by sampling a d-dimensional continuous signal X : [0; 1] → ℝd at a set of times D = {ti}, we construct a piecewise linear, axis-directed process XD : [0, 1] → ℝ2d comprised of a past and future component. We call such an object the Hoff process associated with the discrete data {Xt}ti∈D. The Hoff process can be lifted to its natural rough path enhancement and we consider the question of convergence as the sampling frequency increases. In the specific case where X is a semimartingale we prove that the Itô integral can be recovered from a sequence of random ODEs driven by the components of XD. This is in contrast to the usual Stratonovich integral limit suggested by the classical Wong-Zakai theorem [119]. Such random ODEs have a natural interpretation in the context of mathematical finance. The second part of the thesis introduces a new pathwise approximation scheme for SDEs driven by multidimensional Brownian motion. The work of Cameron, Clarke and Dickinson [17, 37] shows that N-step schemes cannot have a strong approximation error better than O(N-1/2) unless higher iterated integrals of Brownian motion are used. On the other hand, our N-step scheme does not require Lévy area increments and if the vector fields of the SDE possess Stein-Lipschitz regularity of order γ > 2, we prove that the approximation error is O(N-1+ε+2/γ ) for all ε > 0 in the Wasserstein metric from optimal transport theory. By using techniques from rough path theory we avoid imposing any non-degenerate Hörmander or ellipticity conditions on the vector fields of the SDE, which is in contrast to previous pathwise schemes of Alfonsi, Davie, Malliavin et al in [1, 2, 32, 33, 34]. Our scheme is based on the log-ODE method with the Lévy area increments replaced by a quadratic polynomial of Gaussian random variables with the same covariance structure. The coupling construction is an extension of [33, 35] concerning a multidimensional variant of the Komlós-Major-Tusnády theorem and Wasserstein estimates for polynomial perturbations of Gaussian measures. These latter results require no rough path theory and may be of independent interest to probabilists.
Supervisor: Lyons, Terry ; Hambly, Ben Sponsor: European Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
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