We recall that given a $d-$dimensional stochastic process defined as a solution of a homogeneous S.D.E $dX_t = b(X_t)dt + \sigma(X_t)dB_t$ its corresponding infinitesimal generator ${\cal L}$ is s.t it acts on test functions $f$ to give a function ${\cal L}f(x)$ which is, ${\cal L}f(x) = \sum_{i=1}^d b_i(x) \partial_i f + \frac{1}{2} \sum_{i=1,j=1}^d (\sigma \sigma^\top)_{ij} \partial_i \partial_j f$.
Given ${\cal L}$ as above, we define the operator ${\cal L}^*$ s.t for any two bounded measurable $f$ and $h$ we have, $\int ({\cal L}f)h = \int f ({\cal L}^* h)$.
- Can someone kindly give me reference which explains/gives examples as to how is the differential operator form of this ${\cal L}^*$ derived when $\sigma$ is not a constant but has a non-trivial dependence on $X_t$?
The operator ${\cal L}^*$ is important because of the Forward Kolmogorov/Fokker-Plank-Smoluchowski equation which states that if $\rho_t$ is the density of the stochastic process $X_t$ then it satisfies the P.D.E, $\frac{\partial \rho}{\partial t} = {\cal L}^* \rho$. Hence if this P.D.E has to be solved then we necessarily need to find the differential opeartor form of ${\cal L^*}$!
