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.