This talk addresses a problem arising in financial modelling with stochastic differential equations (SDEs). A characterisation theorem is established in which a new link from SDEs to nonlinear parabolic partial differential equations (PDEs) is derived. That is, starting from the necessary and sufficient conditions of the path-independence of the density of Girsanov transform for SDEs, we get a characterisation by means of nonlinear PDEs of Burgers-Kardar-Parisi-Zhang type. Extensions to the cases of degenerated SDEs, jump SDEs, distribution dependent SDEs, as well as to (infinite dimensional) SDEs on separable Hilbert spaces are discussed. Finally, a perspective to stochastically deformed dynamical systems is briefly considered.