Let (f in C^{1,2}left([0, T] times mathbb{R}^{d}, mathbb{R} ight)). We write (partial_{x} f(t, x)) for the row

Let \(f \in C^{1,2}\left([0, T] \times \mathbb{R}^{d}, \mathbb{R}\right)\). We write \(\partial_{x} f(t, x)\) for the row vector \(\left[\frac{\partial f}{\partial x_{i}}(t, x)\right]_{i=1, \ldots, d} ; \partial_{x x} f(t, x)\) for the matrix \(\left[\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}}(t, x)\right]_{i, j}\), and \(\partial_{t} f(t, x)\) for \(\frac{\partial f}{\partial t}(t, x)\). Let \(B=\left(B^{1}, \ldots, B^{n}\right)\) be an \(n\)-dimensional Brownian motion and \(Y_{t}=f\left(t, X_{t}\right)\), where \(X_{t}\) satisfies \(d X_{t}^{i}=\mu_{t}^{i} d t+\sum_{j=1}^{n} \eta_{t}^{i, j} d B_{t}^{j}\). Prove that

\(d Y_{t}=\left\{\partial_{t} f\left(t, X_{t}\right)+\partial_{x} f\left(t, X_{t}\right) \mu_{t}+\frac{1}{2}\left[\eta_{t} \partial_{x x} f\left(t, X_{t}\right) \eta_{t}^{T}\right]\right\} d t+\partial_{x} f\left(t, X_{t}\right) \eta_{t} d B_{t}\).