Stochastic Differential Equations (SDEs)

Physical quantities, such as temperature or pressure, and the functions describing particles and their motion, vary continuously in both space and time. In general, however, noise often makes very jagged functions out of the original smooth functions.

These functions, if noise could be removed, would be smooth by being solutions of equations of motion. Adding intrinsic noise to smooth functions is not an easy task. This has required the development of a new paradigm for Calculus. The Itō calculus and the Malliavin calculus. The former is more popular.

According to Wikipedia, The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates.

There are many tutorials, book chapters, papers, courses content, ..., on this subject. It seems that mathematicians have the most write-ups. Here are some that might be useful.

• Tutorial based on numerical algorithm by Prof. Higham, 2001. Local copy.

• Tutorial on Stochastic Differential Equations by Javier Mogellan.

• An Introduction by Lawrence Evans. that is very mathematically complete but very difficult to read.

• Of course, a visit to Wikipedia is always good!.

• Tutorial 7: A Beginner's Guide to Simulating Stochastic Differential Equations by Jeff Moehlis. Complements Higham's tutorial.

The tutorial by Prof. Higham comes with MATLAB software which can be found here.