March 26th, 2017, 6:41 am
In order to see that underlying of the solution of the Dupire eq could not be the stock one should specified boundary condition for Dupire eq. The boundary condition for the BSE is given at maturity moment T while initial condition for Dupire eq is given at initial moment t = 0. Bearing in mind that Kolmogorov eq is written for initial data of the correspondent stochastic process when time variable is fixed
1) in BSE case we have ( t , S ) are variables and T is fixed we arrive at boundary condition when t = T
2) in Dupire eq ( T , K ) are variables and T = 0 is fixed.
In case 1) underlying is well known stock process S ( T ; t , S ). The variable of the process S ( T ; t , S ) T changes from t to [$] + \infty [$] while parameter t changes from zero to T.
In case 2) underlying is a random process k ( t ; T , K ) . The variable t of the process k ( t ; T , K ) is changes from T to zero, while parameter T is changes from t to [$]+\infty[$].
I did not see that Dupire or others specified the boundary condition for Dupire eq . Though it might be written.
Generally speaking correct probabilistic representation of the BSE and Dupire equations solutions S ( T ; t S ) and k ( t ; T , K ) is a quite sufficient in order never mixed them.
If the fact that the random process can not be associated with the stock it is quite clear that next development of the theory is rather wrong than bad.