The Malliavin Calculus

Acknowledgements I enjoyed the help and support from numerous friends and faculty members during whilst writing this thesis. My greatest debt however, goes to my supervisor, Professor Tony Dooley, who initiated me to the fantastic field of Malliavin calculus. With his incredible breadth and depth of knowledge and intuition, he guided me to structure and clarify my thought and suggested valuable insightful comments. This thesis would not have been possible to write without his help and support. I am also indebted to express my sincere thankfulness to the honours coordinators , Dr Ian Doust and Dr Brian Jefferies, who were very kind and supportive to us during the year. Dr Ben Goldys also deserves a special mention, for the recommendation of a number of useful references, and I very much enjoyed the casual discussions I had with him. Finally, I would like to thank everyone in the School of Mathematics, and the Department of Actuarial Studies for providing me with such a wonderful four years at UNSW. The Malliavin calculus, also known as the stochastic calculus of variations, is an infinite dimensional differential calculus on the Wiener space. Much of the theory builds on from Itô's stochastic calculus, and aims to investigate the structure and also regularity laws of spaces of Wiener functionals. First initiated in 1974, Malli-avin used it in [31] to give a probabilistic proof of the Hörmander's theorem and its importance was immediately recognized. It has been believed up until near the end of the 19th century that a continuous function ought to be smooth at " most points ". The only sort of non-differentiable incidents that were the isolated sharp corners between two pieces of smooth curves, whose behaviour is similar to the graph of f (x) = |x| around x = 0. It was not until 1861, when the German mathematician K. Weierstrass first gave an example of a function that was continuous but nowhere differentiable on R: f (x) = ∞ k=0 3 4 k cos(3 k πx) This was a striking phenomenon at the time, as it signals that there could be a new class of continuous functions that are essentially not governed by almost all of the calculus developed at the time. For example, the integration by substitution formula for gdf breaks down completely if f is nowhere differentiable. This type of wild sharp oscillation is not entirely abstract nonsense. In fact, …