Stochastic flows approach to Dupire's formula

The probabilistic equivalent formulation of Dupire's PDE is the Put-Call duality equality. In local volatility models including exponential L\'{e}vy jumps, we give a direct probabilistic proof for this result based on stochastic flows arguments. This approach also enables us to check the probabilistic equivalent formulation of various generalizations of Dupire's PDE recently obtained by Pironneau by the adjoint equation technique in the case of complex options.