  This paper provides a new formulation of second order stochastic target
problems introduced in [SIAM J. Control Optim. 48 (2009) 2344-2365] by
modifying the reference probability so as to allow for different scales. This
new ingredient enables us to prove a dual formulation of the target problem as
the supremum of the solutions of standard backward stochastic differential
equations. In particular, in the Markov case, the dual problem is known to be
connected to a fully nonlinear, parabolic partial differential equation and
this connection can be viewed as a stochastic representation for all nonlinear,
scalar, second order, parabolic equations with a convex Hessian dependence.
