Brownian motion

Section 1. The de nition and some simple properties. Section 2. Visualizing Brownian motion. Discussion and demysti cation of some strange and scary pathologies. Section 3. The re ection principle. Section 4. Conditional distribution of Brownian motion at some point in time, given observed values at some other times. Section 5. Existence of Brownian motion. How to construct Brownian motion from familiar objects. Section 6. Brownian bridge. Application to testing for unformity. Section 7. A boundary crossing problem solved in two ways: di erential equations and martingales. Section 8. Discussion of some issues about probability spaces and modeling.

Page 5-1

Brownian motion is one of the most famous and fundamental of stochastic processes. The formulation of this process was inspired by the physical phenomenon of Brownian motion, which is the irregular jiggling sort of movement exhibited by a small particle suspended in a uid, named after the botanist Robert Brown who observed and studied it in 1827. A physical explanation of Brownian motion was given by Einstein, who analyzed Brownian motion as the cumulative e ect of innumerable collisions of the suspended particle with the molecules of the uid. Einstein's analysis provided historically important support for the atomic theory of matter, which was still a matter of controversy at the time|shortly after 1900. The mathematical theory of Brownian motion was given a rm foundation by Norbert Wiener in 1923; the mathematical model we will study is also known as the Wiener process." Admittedly, it is possible that you might not share an all-consuming fascination for the motion of tiny particles of pollen in water. However, there probably are any number of things that you do care about that jiggle about randomly. Such phenomena are candidates for modeling via Brownian motion, and the humble Brownian motion process has indeed come to occupy a central role in the theory and applications of stochastic processes. How does it t into the big picture? We have studied Markov processes in discrete time and having a discrete state space. With continuous time and a continuous state space, the prospect arises that a process might have continuous sample paths. To speak a bit roughly for a moment, Markov processes that have continuous sample paths are called di usions. Brownian motion is the simplest di usion, and in fact other di usions can be built up from Brownian motions in various ways. Brownian motion and di usions are used all the time in models in all sorts of elds, such as nance in modeling the prices of stocks, for Stochastic Processes J. Chang, June 1, 1999

Page 5-2

5. BROWNIAN MOTION

example , economics, queueing theory, engineering, and biology. Just as a pollen particle is continually bu eted by collisions with water molecules, the price of a stock is bu eted by the actions of many individual investors. Brownian motion and di usions also arise as approximations for other processes; many processes converge to di usions when looked at in the right way. In fact, in a sense does non-sense count? , Brownian motion is to stochastic processes as the standard normal distribution is to random variables: just as the normal distribution arises as a limit distribution for suitably normalized sequences of random variables, Brownian motion is a limit in distribution of certain suitably normalized sequences of stochastic processes. Roughly speaking, for many processes, if you look at them from far away and not excessively carefully, they look nearly like Brownian motion or di usions, just as the distribution of a sum of many iid random variables looks nearly normal.

5.1 The de nition

Let's scan through the de nition rst, and then come back to explain some of the words in it. 5.1 Definition. A standard Brownian motion SBM fW t : t 0g is a stochastic

process having

i continuous paths, ii stationary, independent increments, and iii W t...