What practical applications do SDEs and SPDEs have?

Question

I will study from the probability thory to its application to stochastic differential equations with my friend. Of cource I'm looking forward to study them but would be a littel discouraging because I don't know their application to practical problems. To say that, I know the fact that they are practicaly used in companies etc. It is that I want to know what kind of scene they are really used in. Although one might say that SDEs or SPDEs are more natural than usual DEs or PDEs, I think that there is no difference between SDEs (SPDEs) and DEs (PDEs) in practical scenes since they are only mathematical subjects. (← This is only my own think!)

I'm glad if you tell me the practicality of SDEs and SPDEs.

And also, if I study while classifying practical application into the field of vision, should I begin study from which of SDEs or SPDEs after understanding basical tools?

Please give me comments.

Answer 1

• In physics of turbulence see "Stochastic PDEs in Turbulence Theory" by E.Weinan.
• In finance see Introduction to Stochastic Differential Equations (SDEs) for Finance
• In biology see "Introduction to Stochastic Models in Biology"

Generally, the main difference with pdes is adding a degenerate process as a potential term that simulates random forcing from the environment. In statistics-ling it represents the sum total of all the confound variables that we can't identify but still affect the system.