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Title: A Stratonovich-Skorohod integral formula for Gaussian rough paths
Author: Lim, Nengli
ISNI:       0000 0004 6348 2406
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2017
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Given a Gaussian process $X$, its canonical geometric rough path lift $\mathbf{X}$, and a solution $Y$ to the rough differential equation $\mathrm{d}Y_{t} = V\left (Y_{t}\right ) \circ \mathrm{d} \mathbf{X}_t$, we present a closed-form correction formula for $\int Y_t \circ \mathrm{d} \mathbf{X}_t - \int Y_t \, \mathrm{d} X_t$, i.e. the difference between the rough and Skorohod integrals of $Y$ with respect to $X$. When $X$ is standard Brownian motion, we recover the classical Stratonovich-to-It{\^o} conversion formula, which we generalize to Gaussian rough paths with finite $p$-variation for $1 \leq p < 3$, and Volterra Gaussian rough paths for $3 \leq p < 4$. This encompasses many familiar examples, including fractional Brownian motion with $H > \frac{1}{4}$. It\^{o}'s formula can also be recovered in the case when $Y_t = \nabla f( X_t)$ for some smooth $f$. \par To prove the formula, we show that $\int Y_t \, \mathrm{d} X_t$ is the $L^2(\Omega)$ limit of its Riemann-sum approximants, and that the approximants can be appended with a suitable compensation term without altering the limit. To show convergence of the Riemann-sum approximants, we utilize a novel characterization of the Cameron-Martin norm using multi-dimensional Young-Stieltjes integrals. For the main theorem, complementary regularity between the Cameron-Martin paths and the covariance function of $X$ is used to show the existence of these integrals. However, it turns out not to be a necessary condition, as we provide a new set of conditions for their existence, as well as provide a new formulation of the classical It\^{o}-Skorohod isometry in terms of these Young-Stieltjes integrals.
Supervisor: Cass, Thomas Sponsor: Imperial College London
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral