  We introduce an extension of Dual Dynamic Programming (DDP) to solve convex
nonlinear dynamic programming equations. We call Inexact DDP (IDDP) this
extension which applies to situations where some or all primal and dual
subproblems to be solved along the iterations of the method are solved with a
bounded error. We show that any accumulation point of the sequence of decisions
is an approximate solution to the dynamic programming equations. When these
errors tend to zero as the number of iterations goes to infinity, we show that
IDDP solves the dynamic programming equations. We extend the analysis to
stochastic convex nonlinear dynamic programming equations, introducing Inexact
Stochastic Dual Dynamic Programming (ISDDP), an inexact variant of SDDP
corresponding to the situation where some or all problems to be solved in the
forward and backward passes of SDDP are solved approximately. We also show the
almost sure convergence of ISDDP for vanishing errors.
