Strong solutions of SDEs in electrical engineering

Question

I am currently reading about existence and uniqueness theory for stochastic differential equations (SDE). Two of the main concepts are: strong and weak solutions.

I do understand the difference between the two (for strong solutions we have a given Brownian motion on a given probability space, whereas for weak solutions we get to choose both, the Brownian motion and the probability space).

Now I am interested in real world applications. I remember vaguely that strong solutions are very important in certain applications in signal processing/EE. It kind of makes sense to me (as a mathematician), that in signal processing applications the notion of strong solution is very important (we can not choose the noise ourselves).

I found the following passage in the book Stochastic Calculus: A Practical Introduction by Rick Durrett:

In signal processing applications one must deal with the noisy signal that one is given, so strong solutions are required. However, if the aim is to construct diffusion processes then there is nothing wrong with a weak solution.

Can anyone help me make this more precise? Which applications in EE require strong solutions? Are there any applications in EE in which weak solutions are enough?

Any help is appreciated.

Answer 1

The book by Karatzas & Shreve is a good introduction into the essential stuff about strong and weak solutions. In short:

• you are correct that a weak solution gives us the freedom to choose the noise ourselves.

• weak solutions of $dX_t=\sigma(t,X_t)\,dW_t+b(t,X_t)\,dt$ exist under much milder conditions on the coefficient functions $\sigma(t,x),b(t,x)\,.$

• if you are only interested in the distribution of $X$ a weak solution will do.

• an important result by Yamada & Watanabe says: weak existensc plus pathwise uniqueness implies strong existence. This means that it can sometimes help to look for a weak solution first.

• I do not think that there will be a one-to-one map from a real world problem to a particular notion of SDE solution. As always it is judgement combined with doing what is practical.