It is well known that if $X_t$ is a stochastic process that solves the SDE $$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$ with $W_t$ a Wiener process, then the associated probability density $p(x,t)$ solves $$\frac{\partial p}{\partial t} = -\nabla \cdot (\mu p) + \frac 12 \nabla \nabla : (Dp),$$ where $D_{ij} =\sum_{k} \sigma_{ik}\sigma_{jk}$ is symmetric, positive semi-definite. (as summarized e.g. here)
Now, in the same article, it is written that by interpreting the stochastic SDE in the Stratonovich sense rather than the Ito sense, one instead arrives at the corresponding equation $$\frac{\partial p}{\partial t} = -\nabla \cdot (\mu p) + \frac 12 \sum_{i,k} \frac{\partial}{\partial x_i}\left( \sigma_{ik} \sum_{j}\frac{\partial}{\partial x_j}\left[\sigma_{kj} p\right]\right).$$ I'm wondering if there is yet a third way to interpret the SDE which would lead to a parabolic equation in divergence-form for the probability density? I.e. to $$\frac{\partial p}{\partial t} = -\nabla \cdot (\mu p) + \frac 12 \sum_{i,j} \frac{\partial}{\partial x_i}\left( D_{ij} \frac{\partial p}{\partial x_j}\right).$$ The motivation for the last form is that it seems to arise very natural in physics. Does anyone know if that's possible? Thank you.
