These are the lecture notes for a one quarter graduate course in Stochastic Processes that I taught at Stanford University in 2002 and 2003. This course is intended
for incoming master students in Stanford’s Financial Mathematics program, for advanced undergraduates majoring in mathematics and for graduate students from
Engineering, Economics, Statistics or the Business school. One purpose of this text
is to prepare students to a rigorous study of Stochastic Differential Equations. More
broadly, its goal is to help the reader understand the basic concepts of measure theory that are relevant to the mathematical theory of probability and how they apply
to the rigorous construction of the most fundamental classes of stochastic processes.
Towards this goal, we introduce in Chapter 1 the relevant elements from measure
and integration theory, namely, the probability space and the σ-fields of events
in it, random variables viewed as measurable functions, their expectation as the
corresponding Lebesgue integral, independence, distribution and various notions of
convergence. This is supplemented in Chapter 2 by the study of the conditional
expectation, viewed as a random variable defined via the theory of orthogonal
projections in Hilbert spaces.
After this exploration of the foundations of Probability Theory, we turn in Chapter
3 to the general theory of Stochastic Processes, with an eye towards processes
indexed by continuous time parameter such as the Brownian motion of Chapter
5 and the Markov jump processes of Chapter 6. Having this in mind, Chapter
3 is about the finite dimensional distributions and their relation to sample path
continuity. Along the way we also introduce the concepts of stationary and Gaussian
Chapter 4 deals with filtrations, the mathematical notion of information progression in time, and with the associated collection of stochastic processes called
martingales. We treat both discrete and continuous time settings, emphasizing the
importance of right-continuity of the sample path and filtration in the latter case.
Martingale representations are explored, as well as maximal inequalities, convergence theorems and applications to the study of stopping times and to extinction
of branching processes.
Chapter 5 provides an introduction to the beautiful theory of the Brownian motion. It is rigorously constructed here via Hilbert space theory and shown to be a
Gaussian martingale process of stationary independent increments, with continuous
sample path and possessing the strong Markov property. Few of the many explicit
computations known for this process are also demonstrated, mostly in the context
of hitting times, running maxima and sample path smoothness and regularity.