Affine Transformations of It Diffusions and their Transition Densities

For a given It diffusion, we derive the forward Kolmogorov equation (FKE) associated with the adjoint operator of the infinitesimal generator of an affine transformation of the given It diffusion. The fundamental solution obtained by solving the FKE is, in fact, the transition density of the transformed diffusion. Moreover, we prove that the transition density can be represented in terms of a product of two functions, a Jacobian term and a composition of the transition density of the given It diffusion and the inverse of the transformation. Finally, we present an application of our results in parameter estimation in commodity markets in which the commodity prices are assumed to follow an extended Black-Scholes model.