Diffusions, Markov Processes and Martingales

Abstract

This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Reflected planar Brownian motions, intertwining relations and crossing probabilities

“…[RevYor91]), and that the harmonic measure on the real line seen from a + ib is given by a Cauchy law (see e.g. [RogWil93]) with density:…”

Discounted optimal stopping problems for the maximum process

“…An example (the Russian option) of problem (1.2) was solved in the framework of option pricing theory by Shepp & Shiryaev [9], [10], when the diffusion (X t ) is a geometric Brownian motion, λ(x) is a positive constant and D(x) ≡ 0 (see also [1] and [3]). Let ( X t ) be the killed diffusion at rate λ(·) of (X t ) [8]. If a new point Δ is adjoined to the state space I = [0, ∞), and we set I Δ = [0, ∞) ∪ {Δ}, the (homogeneous) transition function of the process ( X t ) is given by…”

Diffusion Maps for dimensionality reduction and visualization of meteorological data

“…We denote by π the stationary distribution of this Markov process [18], which is given by π(x) = gα(xi) j gα(xj ) . We can now consider one-step Markov neighborhoods or, more generally, larger t-step neighborhoods, i.e., those points accessible from a given x i in t Markov steps.…”