  In this paper, we consider multi-stage stochastic optimization problems with
convex objectives and conic constraints at each stage. We present a new
stochastic first-order method, namely the dynamic stochastic approximation
(DSA) algorithm, for solving these types of stochastic optimization problems.
We show that DSA can achieve an optimal ${\cal O}(1/\epsilon^4)$ rate of
convergence in terms of the total number of required scenarios when applied to
a three-stage stochastic optimization problem. We further show that this rate
of convergence can be improved to ${\cal O}(1/\epsilon^2)$ when the objective
function is strongly convex. We also discuss variants of DSA for solving more
general multi-stage stochastic optimization problems with the number of stages
$T > 3$. The developed DSA algorithms only need to go through the scenario tree
once in order to compute an $\epsilon$-solution of the multi-stage stochastic
optimization problem. As a result, the memory required by DSA only grows
linearly with respect to the number of stages. To the best of our knowledge,
this is the first time that stochastic approximation type methods are
generalized for multi-stage stochastic optimization with $T \ge 3$.
