Fokker-Planck Equation for Systems of SDEs

We describe the generalization of the Ito-Doeblin formula and the Fokker-Planck equation to a system of SDEs. If is a scalar valued function of M , and Z(f) is the solution of the SDE system 

All components on the right hand side of (6.40) are evaluated at Z(f). Then using the shorthand rules dW,dW = dtSij (where Sij is one if i = j, zero otherwise), we have 

In the multivariate case, the density is a function of the vector describing a state of the system z and t, and is governed by the partial differential equation 

The second order diffusion term in this expression may be expanded as 

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