Let (X) be a semi-martingale such that (dlangle Xangle_{t}=sigma^{2}left(t, X_{t} ight) d t). Assuming that the law

Let \(X\) be a semi-martingale such that \(d\langle Xangle_{t}=\sigma^{2}\left(t, X_{t}\right) d t\). Assuming that the law of the r.v. \(X_{t}\) admits a density \(\varphi(t, x)\), prove that, under some regularity assumptions,

\[\mathbb{E}\left(d_{t} L_{t}^{x}\right)=\varphi(t, x) \sigma^{2}(t, x) d t .\]