  We aim to provide a Feynman-Kac type representation for
Hamilton-Jacobi-Bellman equation, in terms of forward backward stochastic
differential equation (FBSDE) with a simulatable forward process. For this
purpose, we introduce a class of BSDE where the jumps component of the solution
is subject to a partial nonpositive constraint. Existence and approximation of
a unique minimal solution is proved by a penalization method under mild
assumptions. We then show how minimal solution to this BSDE class provides a
new probabilistic representation for nonlinear integro-partial differential
equations (IPDEs) of Hamilton-Jacobi-Bellman (HJB) type, when considering a
regime switching forward SDE in a Markovian framework, and importantly we do
not make any ellipticity condition. Moreover, we state a dual formula of this
BSDE minimal solution involving equivalent change of probability measures. This
gives in particular an original representation for value functions of
stochastic control problems including controlled diffusion coefficient.
