  In 2002, Benjamin Jourdain and Claude Martini discovered that for a class of
payoff functions, the pricing problem for American options can be reduced to
pricing of European options for an appropriately associated payoff, all within
a Black-Scholes framework. This discovery has been investigated in great detail
by S\"oren Christensen, Jan Kallsen and Matthias Lenga in a recent work in
2020. In the present work we prove that this phenomenon can be observed in a
wider context, and even holds true in a setup of non-linear stochastic
processes. We analyse this problem from both probabilistic and analytic
viewpoints. In the classical situation, Jourdain and Martini used this method
to approximate prices of American put options. The broader applicability now
potentially covers non-linear frameworks such as model uncertainty and
controller-and-stopper-games.
