It’s all nice, but can we do anything else with it?
Sure we can! I must warn you though, things start to get complicated and technical, so for those that are willing to hear a bunch of terms and see a couple of definition, this will be all there is to it!
· Stochastic Integral
Sometimes it is useful to define integrals over the increments of Brownian motion, that is quantities of the form .
Here f(t,ù) is a random variable and dBt is the increment of the Brownian motion in the interval from t to t+dt.
Unfortunately this integrals inherits the pathogenic behavior of Brownian motion, so there is a need for a new definition
The problem was solved by Ito, who defined what is today known as “Ito integral”, which follows a somewhat strange calculus, as compared to the usual Riemann integral.
As an example, using Ito’s integral we will get ,
where the second term shows the difference from what we would expect from our usual rules of integration!
· Stochastic differential equations
A stochastic differential equation is an equation of the form
Or in integral form , where Bt is a Brownian motion.
The two hypotheses we have to make for b and ó are the following
An important example is the Ornstein-Uhlenbeck process , which has important applications in physics, as a particle subject to a stochastic force and a friction analogous to is velocity.
However, the whole structure emanating from Brownian motion is not restricted to physics, but is also extended to other areas, from biology to finance!
Food for thought!
For those who didn’t suffer enough, here are two books on the subject:
“Basic Stochastic Processes” by Brzezniak and Zastawniak
“Stochastic Differential Equations” by B. Oksendal (more mathematically challenging than the above)
Is there anything left?
Of course! The title is “Plasmas and Random Walks”. So far we talked a bit about both. But the connection remains unseen!
As a closing remark, I will present VERY briefly one connection between the two.