# Connections between random walk and heat equation (Material for ~)

I am preparing an undergraduate lecture in quantitative finance and I am looking for material that combines the topics:

The material should be accessible (intuitive!), give some background (so not only proving that the random walk is the solution to the heat equation) and could also address adjacent and/or supporting topics.

My question
Could you provide me with references, links etc.?

• Interesting - using the heat equation as setup for Black-Scholes? – user3137 Oct 22 '12 at 18:24
• @x711Li: Yes, this is one of the routes I want to take. – vonjd Oct 22 '12 at 18:50
• Einstein's derivation of the diffusion equation is really intuitive. I can't find a good ref right now... – Ryogi Oct 22 '12 at 22:38

## 1 Answer

I would start with explaining random walk (this should be fairly simple) and then making a connection to heat equation in discrete time. This paper is doing exactly this and by leaving out technicalities you should make this pretty intuitive for students.

Basically the intuition is as follows:

At each integer time unit, the heat at each point is spread evenly among its nearest neighbors. If one of those neighbors is a boundary point, then the heat that goes to that site is lost forever.

Probabilistic view of the heat is given by imagining that the temperature is controlled by a very large number of “heat particles”. These particles perform random walks until they leave the object at which time they are killed. The temperature at each point and time, is given by the density of particles at this point.