Ito’s formula, which characterises the stochastic differential, has been mentioned by various textbooks and courses, but now for the first time (after James Norris’s first lecture for the Stochastic Calculus course) I think I finally have a reasonable idea of what’s going on. The reasons I was initially confused help to explain what the motivation is:
- What processes can we consider? Well, initially continuous time, time-homogeneous Markov processes in with continuous paths. It could be space-homogeneous as well if desired. By the theory of decomposition of Levy processes (ie what we are considering), the continuous paths property gives that such a process must be a Brownian motion with drift. This has the property that where is the diffusivity, that is, the intensity of the Brownian component, and is the drift.
- What is the stochastic differential? Well, for a process as above, we define: . This is non-deterministic: that’s reasonable since X is a stochastic process. And, a normal differential is meaningful only when you integrate, so similarly the stochastic differential is only meaningful when you take an expectation.
- Write for the Brownian noise. Then , so by Taylor: , remembering that .
- This is generally written as where . Now note that and , so it is reasonable that we might ‘cancel the expectations’ to get: .
- Use a suitable tensor product or when d>1.
- This is (a version of) Ito’s Lemma.