### Topics from 2001

**Conditioning**
- elementary case
- conditional distributions (postpone until discussion of BM)
- fair price interpretation
- the abstract Kolmogorov conditional expectation; sigma-fields as
``information"; projection interpretation

**Stochastic processes**
- filtrations
- optional times (a.k.a. stopping times)
- sample path properties
- versions of processes and the "usual conditions"

**Martingales**
- a quick overview of theory for discrete-time martingales,
as treated in Statistics 600
- uniform integrability
- sample path properties of martingales in continuous time: existence of cadlag versions
- preservation of martingale properties at stopping times
- the martingale central limit theorem, as an introduction to the role of
quadratic variation

**Brownian motion**
- construction of a Brownian motion with continuous sample paths
- Lévy's martingale characterization of Brownian motion
- Itô integral as the prime example of a stochastic integral
with respect to a (locally) square integrable martingale

**Predictability** (omit some parts if short of time)
- discussion of the subtle connections between prediction of processes and
measurability with respect to the predictable sigma-field
- statement and explanation of the section and projection theorems (maybe)
- foretelling of predictable stopping times
- characterization of predictable processes; predictable projections
- predictable measures and predictable increasing processes
- compensators and the Doob-Meyer decomposition of a submartingale
- maybe something on point processes

**Stochastic integral with respect to a square integrable martingale**
- the Doléans measure for a submartingale
- the stochastic integral via the L
^{2}(Doléans) isometry

**Localization and semimartingales**
- local martingales etc
- stochastic integral with respect to a semimartingale
- quadratic variation [square-brackets process]
- Itô's formula for semimartingales

**Change of measure**
- Black-Scholes formula via change of measure

**Diffusions and stochastic differential equations** (maybe)

www.stat.yale.edu/~pollard/603.fall04

30 August 2004