Prove the following time-dependent version of It's formula: let (left(B_{t}ight)_{t geqslant 0}) be a (mathrm{BM}^{1}) and (f:[0,

Prove the following time-dependent version of Itô's formula: let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(f:[0, \infty) \times \mathbb{R} ightarrow \mathbb{R}\) be a function of class \(\mathcal{C}^{1,2}\). Then

\[f\left(t, B_{t}ight)-f(0,0)=\int_{0}^{t} \frac{\partial f}{\partial x}\left(s, B_{s}ight) d B_{s}+\int_{0}^{t}\left(\frac{\partial f}{\partial t}\left(s, B_{s}ight)+\frac{1}{2} \frac{\partial^{2} f}{\partial x^{2}}\left(s, B_{s}ight)ight) d s .\]

Prove and state the \(d\)-dimensional counterpart.

Use the Itô formula for the \((d+1)\)-dimensional Itô process \(\left(t, B_{t}^{1}, \ldots, B_{t}^{d}ight)\).