During my last exam session in Maynooth I found myself not wanting to study for my exams. Naturally, the thing to do here is to go down some completely unrelated rabbit holes, which serve no purpose for my exams. One of these rabbit holes was a discovery, to me, of the subject of stochastic calculus. A strange but familiar looking cousin to the calculus we all know, but with its own quirks and very much different motivations. These motivations, as opposed to the ones of the classical calculus which grew from the need to describe physical systems, grew to describe what has now become a pillar of the field of finiancial mathematics. In essence, it helps to describe stock market prices. I will ty here to draw a parallel bewteen two theorems, one of calculus and one of this stochastic calculus. First I'll state the two theorms.

Theorem (Ito's Formula simpest case)

If $B_t$ is the standard brownian motion and $ f:\mathbb{R} \to \mathbb{R} $ has a continious second derivative then $$ f(B_t) = f(0) + \int_0^t f'(B_s) dB_s + \frac{1}{2} \int_0^t f''(B_s) ds \hspace{4cm} (1)$$ And if we rearange a little bit we get $$ \int_0^t f'(B_s) dB_s + \frac{1}{2} \int_0^t f''(B_s) ds = f(B_t) - f(0) \hspace{4cm} (2)$$

Theorem (Fundamental Theorem of calculus second part)

Let $f:[a,b] \to \mathbb{R}$. If $f$ is Riemann integrable on $[a,b]$ then $$ \int_a^b f'(x) dx = f(b) - f(a) \hspace{4cm} (3)$$ And to try to match the notation in the previous theorem by a substitution or two we get $$ \int_0^t f'(x) dx = f(t) - f(0) \hspace{4cm} (4)$$ Now the key comparison is to make is bewteen (2) and (4). We can note that in both equations we share an integral whose integrand is a derivative of our function and a difference of our function $f$ evaluated at $t$ and $0$, but in (2) there is an extra integral term. I think this is a kinda unexpected, wierd result but it is a sort of alternative calculus to what most of us know which I think makes this really interesting.