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Stochastic Differential Equations

Introduction of the course. Brownian motion was named after English botanist Robert Brown who observed that pollen grains moved randomly in a drop of water in 1827. In 1905 Albert Einstein related the Brownian motion to a diffusion equation, also known as ``heat equation.'' He considered the density $ p(x,t)$ of particles at point $ x$ at time $ t$ , and derive the normal distribution as the solution to the corresponding heat equation. Later in 1920's Norbert Wiener described the Brownian motion on a firm mathematical theory, now called a ``Wiener process.''

Explore the Brownian motion. A demonstration

BS.R

was written by programming language R, and downloaded from our course web site. You also need to install R which is available free from

CRAN R project

Then start R, and open the source code BS.R by the menu [File]->[Source R code...] at the system.

Choose the first topic ``brownian.motion,'' and adjust the window so that the presentation material is appropriately displayed. To start the demonstration, click the mouse on the top of [Demo]. By clicking the mouse anywhere above [Probability Density] you can see the density curve at a particular time. Move the mouse to the right side of the vertical line above [Run], and click the mouse to see a sample path of Brownian motion. Click again and again, you will see the behavior of randomly generated sample paths.

Is finance a science? Black, Merton and Scholes developed a pioneering formula for option pricing in 70's. We will looked at their treatise of European call option, hedged position, and portfolio strategy all described in terms of stochastic differential equations. Thus, the part of our course may be viewed as an introduction to mathematical finance. Watch the documentary whose location is linked to the course web site. It is a very good introduction to the little-known history of predicting financial markets, including the interviews with some successful modern traders who rely on intuition as well as mathematical models.

Part 1; Part 2; Part 3; Part 4; Part 5

Online lecture notes. We frequently refer to Evans' lecture note for actual definitions and important results. In our short course we quickly move to the foundation of stochastic differential equations (SDE's) in Chapters 4 and 5, but omit most of the content involving actual proofs. For our objective of understanding the SDE's, we consider our coverage of examples in Chapter 5 as the centerpiece of these two chapters.

Tentative schedule. Lectures are problem-based, and our goal is to complete all the problems presented in the class.

  1. Basic concept in probability theory, and an introduction to Brownian motion. Problem set #1: Heat equation and Brownian motion.
  2. An introduction to SDE and Ito calculus. Problem set #2: Applications of Ito formula and SDE's
  3. More example of SDE's and their solutions.
  4. An introduction to mathematical finance and option pricing. Problem set #3: Deriving the Black-Scholes formula.
  5. Further discussion on applications of SDE to mathematical finance.
  6. An introduction to filtering problems and the mechanism of Kalman-Bucy filter. Problem set #4: Solving linear filtering problems.
  7. Further discussion on filtering problems: Girsanov's theorem and Bayes formula.


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