Abstract For a one-dimensional Ito process X t = ∫ 0 t σ s d W s and a general F t X -adapted non-decreasing path-dependent functional Y t , we derive a number of forward equations for the characteristic function of ( X t , Y t ) for absolutely and non absolutely continuous functionals Y t . The functional Y t can be the maximum, the minimum, the local time, the quadratic variation, the occupation time or a general additive functional of X . Inverting the forward equation, we obtain a new Fourier-based method for computing the Markovian projection E ( σ t 2 | X t , Y t ) explicitly from the marginals of ( X t , Y t ) , which can be viewed as a natural extension of the Dupire formula for local volatility models; E ( σ t 2 | X t , Y t ) is a fundamental quantity in the important mimicking theorems in Brunick and Shreve (2013). We also establish mimicking theorems for the case when Y is the local time or the quadratic variation of X (which is not covered by Brunick and Shreve (2013)), and we derive similar results for trivariate Markovian projections.