Ito Calculus

Formalization of the Theory of integration applied to Random variables. More specifically, we are able to calculate $Y_t={\int }_0^t{H}_{s}d{X}_s$ where:

• $X_s$ is a Semimartingale. I believe this is a Wiener Process with jumps, but I am unsure. Help me ❓
• $H_s$ is adapted to the Filtration generated by $X_s$. More specifically, $H_t$ is a measurable function on $\sigma (\{X_s:s\le t\})$.

260 page Intro by Gregory F. Lawler

• Describe the Strotonovich Integral , and compare the pros/cons with the Ito’s definition of the integral. One uses the left limit when defining $d{W}_t$, whereas the other takes the central value. See https://oatml.cs.ox.ac.uk/blog/2022/03/22/ito-strat.html for a good overview of the differences