"In a first course on probability one typically works with a sequence of random variables X1,X2,... For stochastic processes, instead of indexing the random variables by the non-negative integers, we index them by t G [0, oo) and we think of Xt as being the value at time t. The random variable could be the location of a particle on the real line, the strength of a signal, the price of a stock, and many other possibilities as well. We will also work with increasing families of s -fields {J-t}, known as filtrations. The s -field J-t is supposed to represent what we know up to time t. 1.1 Processes and s -fields Let (Q., J-, P) be a probability space. A real-valued stochastic process (or simply a process) is a map X from [0, oo) x Q. to the reals. We write Xt = Xt(?) = X(t, ?). We will impose stronger measurability conditions shortly, but for now we require that the random variables Xt be measurable with respect to J- for each t 0. A collection of s -fields J-t such that J-t C J- for each t and J-s C J-t ifs t is called a filtration. Define J-t+ = P\e0J-t+e. A filtration is right continuous if J-t+ = J-t for all t 0. "--

Preface 1 Basic notions 2 Brownian motion 3 Martingales 4 Markov properties of Brownian motion 5 The Poisson process 6 Construction of Brownian motion 7 Path properties of Brownian motion 8 The continuity of paths 9 Continuous semimartingales 10 Stochastic integrals 11 Itô's formula 12 Some applications of Itô's formula 13 The Girsanov theorem 14 Local times 15 Skorokhod embedding 16 The general theory of processes 17 Processes with jumps 18 Poisson point processes 19 Framework for Markov processes 20 Markov properties 21 Applications of the Markov properties 22 Transformations of Markov processes 23 Optimal stopping 24 Stochastic differential equations 25 Weak solutions of SDEs 26 The Ray-Knight theorems 27 Brownian excursions 28 Financial mathematics 29 Filtering 30 Convergence of probability measures 31 Skorokhod representation 32 The space C[0, 1] 33 Gaussian processes 34 The space D[0, 1] 35 Applications of weak convergence 36 Semigroups 37 Infinitesimal generators 38 Dirichlet forms 39 Markov processes and SDEs 40 Solving partial differential equations 41 One-dimensional diffusions 42 Lévy processes A Basic probability B Some results from analysis C Regular conditional probabilities D Kolmogorov extension theorem E Choquet capacities References Index