Clarification about a very simple stochastic integral

Question

I'm studying stochastic integrals right now and I feel like this question is incredibly easy but I'm not sure. I want to evaluate $\int_0^t sdB_s$. Using Ito's formula I get $tB_t$ by setting $f(B_s)=sB_s$ in the formula but I'm wondering if this is correct or should I treat the s like a constant or what? Thanks for any help.

Answer 1

Since the function $f$ depends on the time $t$, we have to apply the time-dependent Itô formula which states that

$$f(t,B_t)-f(0,B_0) = \int_0^t \frac{\partial}{\partial x} f(s,B_s) \, dB_s + \int_0^t \left( \frac{\partial}{\partial t}f(s,B_s) + \frac{1}{2} \frac{\partial^2}{\partial x^2} f(s,B_s) \right) \, ds.$$

For $f(t,x) := t \cdot x$ we see that

$$t \cdot B_t - 0 = \int_0^t s \, dB_s + \int_0^t (B_s+0) \, ds,$$

i.e.

$$\int_0^t s \, dB_s = t \cdot B_t - \int_0^t B_s \, ds.$$