  The paper studies a non-linear transformation between Brownian martingales,
which is given by the inverse of the pricing operator in the mathematical
finance terminology. Subsequently, the solvability of systems of equations
corresponding to such transformations is investigated. The latter give rise to
novel monotone pathwise couplings of an arbitrary number of certain diffusion
processes with varying diffusion coefficients. In the case that there is an
uncountable number of these diffusion processes and that the index set is an
interval such couplings can be viewed as models for the growth of
one-dimensional random surfaces. With this motivation in mind, we derive the
appropriate stochastic partial differential equations for the growth of such
surfaces.
