  Least Squares regression was first introduced for the pricing of
American-style options, but it has since been expanded to include swing options
pricing. The swing options price may be viewed as a solution to a Backward
Dynamic Programming Principle, which involves a conditional expectation known
as the continuation value. The approximation of the continuation value using
least squares regression involves two levels of approximation. First, the
continuation value is replaced by an orthogonal projection over a subspace
spanned by a finite set of $m$ squared-integrable functions (regression
functions) yielding a first approximation $V^m$ of the swing value function. In
this paper, we prove that, with well-chosen regression functions, $V^m$
converges to the swing actual price $V$ as $m \to + \infty$. A similar result
is proved when the regression functions are replaced by neural networks. For
both methods (least squares or neural networks), we analyze the second level of
approximation involving practical computation of the swing price using Monte
Carlo simulations and yielding an approximation $V^{m, N}$ (where $N$ denotes
the Monte Carlo sample size). Especially, we prove that $V^{m, N} \to V^m$ as
$N \to + \infty$ for both methods and using Hilbert basis in the least squares
regression. Besides, a convergence rate of order
$\mathcal{O}\big(\frac{1}{\sqrt{N}} \big)$ is proved in the least squares case.
Several convergence results in this paper are based on the continuity of the
swing value function with respect to cumulative consumption, which is also
proved in the paper and has not been yet explored in the literature before for
the best of our knowledge.
