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Title: | A STRATONOVICH - SKOROHOD INTEGRAL FORMULA FOR GAUSSIAN ROUGH PATHS |

Authors: | LIM NENG-LI |

Keywords: | Rough paths, Gaussian processes, fractional Brownian motion, Stratonovich integration, Skorohod integration |

Issue Date: | 20-Jan-2017 |

Citation: | LIM NENG-LI (2017-01-20). A STRATONOVICH - SKOROHOD INTEGRAL FORMULA FOR GAUSSIAN ROUGH PATHS. ScholarBank@NUS Repository. |

Abstract: | 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. |

URI: | http://scholarbank.nus.edu.sg/handle/10635/134935 |

Appears in Collections: | Ph.D Theses (Open) |

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Nengli Lim Thesis 2016.pdf | 708.2 kB | Adobe PDF | OPEN | None | View/Download |

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