Self-Study material - Stochastic Differential Equations

The lecture follows Chapters 3.1-3.5 in Åström Introduction to Stochastic Control Theory, see Material

You might find this video showing some examples of stochastic integrals in Mathematica illustrative.

Or this video about Ito's lemma

A detailed, rigorous definition of SDEs, proofs of Ito's lemma, and more advanced analysis of SDEs can be found in this online textbook by A. Eberle, Univ. of Bonn

In the splitting of the spectral distribution function F into three different parts, as mentioned in the lecture

$F (ω) = F_{a} (ω) + F_{d} (ω) + F_{s} (ω)$

the so-called singular measure $F_{s} (ω)$ is hard to understand, but is related to existence of strange mathematical objects such as the Cantor function and the Cantor set.