On the Markovian projection in the Brunick-Shreve mimicking result

For a one-dimensional Itô process Xt=∫0tσsdWs and a general FtX-adapted non-decreasing path-dependent functional Yt, we derive a number of forward equations for the characteristic function of (Xt,Yt) for absolutely and non absolutely continuous functionals Yt. The functional Yt 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(σt2|Xt,Yt) explicitly from the marginals of (Xt,Yt), which can be viewed as a natural extension of the Dupire formula for local volatility models; E(σt2|Xt,Yt) 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.