On Higher Derivatives of Expectations

It is understood that derivatives of an expectation $E [\phi(S(T)) | S(0) = x]$ with respect to $x$ can be expressed as $E [\phi(S(T)) \pi | S(0) = x]$, where $S(T)$ is a stochastic variable at time $T$ and $\pi$ is a stochastic weighting function (weight) independent of the form of $\phi$. Derivatives of expectations of this form are encountered in various fields of knowledge. We establish two results for weights of higher order derivatives under the dynamics given by (\ref{dynamics}). Specifically, we derive and solve a recursive relationship for generating weights. This results in a tractable formula for weights of any order.