Title:

Discrete approximations in stochastic rough path theory

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 leadlag increments. In particular, by sampling a ddimensional continuous signal X : [0; 1] → ℝ^{d} at a set of times D = {t_{i}}, we construct a piecewise linear, axisdirected process X^{D} : [0, 1] → ℝ^{2d} comprised of a past and future component. We call such an object the Hoff process associated with the discrete data {X_{t}}_{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 X^{D}. This is in contrast to the usual Stratonovich integral limit suggested by the classical WongZakai 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 Nstep 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 Nstep scheme does not require Lévy area increments and if the vector fields of the SDE possess SteinLipschitz 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 nondegenerate 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 logODE 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 KomlósMajorTusnády 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.
