  We introduce an extension of Stochastic Dual Dynamic Programming (SDDP) to
solve stochastic convex dynamic programming equations. This extension applies
when some or all primal and dual subproblems to be solved along the forward and
backward passes of the method are solved with bounded errors (inexactly). This
inexact variant of SDDP is described both for linear problems (the
corresponding variant being denoted by ISDDP-LP) and nonlinear problems (the
corresponding variant being denoted by ISDDP-NLP). We prove convergence
theorems for ISDDP-LP and ISDDP-NLP both for bounded and asymptotically
vanishing errors. Finally, we present the results of numerical experiments
comparing SDDP and ISDDP-LP on portfolio problem with direct transaction costs
modelled as a multistage stochastic linear optimization problem. On these
experiments, ISDDP-LP allows us to obtain a good policy faster than SDDP.
