Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$
- $(B_t)_{t\ge 0}$ be a $\mathbb R^d$-valued Brownian motion with respect to $\mathcal F$
Let $v(t,X_t)$ be the velocity of a particle $X_t\in\mathbb R^d$ at time $t\ge 0$. Assuming $v:[0,\infty)\times\mathbb R^d\to\mathbb R^d$ is sufficiently smooth, its material derivative is given by $${\rm D}v:=\partial_tv+v\cdot\nabla v\tag 1\;.$$ If $t\mapsto X_t$ is the trajectory of a particle, we obtain $$X_t=X_0+\int_0^tv_s(X_s)\,{\rm d}s\;.\tag 2$$
Assuming that the motion of the particle is perturbed by a random forcing, we may replace $(2)$ by $$X_t=X_0+\int_0^tv_s(X_s)\,{\rm d}s+\int_0^t\xi_s(X_s)\,{\rm d}B_s\tag 3$$ for some $\xi:[0,\infty)\times\mathbb R^d\to\mathbb R^{d\times d}$. How can we evaluate $(1)$ in this case by means of an Itō formula (instead of applying the chain rule)?