Inflation Derivatives and European Central Bank

European inflation and interest rates are closed related. In particular we consider the relationship among European inflation, European Central Bank official interest rate and short-term interest rate in a stochastic continuous time setting for the valuation of inflation derivatives. We model the state variables in two different time scales and we provide some results about the existence and uniqueness of: (i) a strong solution of a more general jump-diffusion problem in which our model is included, and (ii) a solution, in the classical sense, of the valuation equation as a series of solutions of degenerate parabolic PDEs with non-local terms.