  We consider a class of sampling-based decomposition methods to solve
risk-averse multistage stochastic convex programs. We prove a formula for the
computation of the cuts necessary to build the outer linearizations of the
recourse functions. This formula can be used to obtain an efficient
implementation of Stochastic Dual Dynamic Programming applied to convex
nonlinear problems. We prove the almost sure convergence of these decomposition
methods when the relatively complete recourse assumption holds. We also prove
the almost sure convergence of these algorithms when applied to risk-averse
multistage stochastic linear programs that do not satisfy the relatively
complete recourse assumption. The analysis is first done assuming the
underlying stochastic process is interstage independent and discrete, with a
finite set of possible realizations at each stage. We then indicate two ways of
extending the methods and convergence analysis to the case when the process is
interstage dependent.
