  This paper is concerned with optimal switching over multiple modes in
continuous time and on a finite horizon. The performance index includes a
running reward, terminal reward and switching costs that can belong to a large
class of stochastic processes. Particularly, the switching costs are modelled
by right-continuous with left-limits processes that are quasi-left-continuous
and can take both positive and negative values. We provide sufficient
conditions leading to a well known probabilistic representation of the value
function for the switching problem in terms of interconnected Snell envelopes.
We also prove the existence of an optimal strategy within a suitable class of
admissible controls, defined iteratively in terms of the Snell envelope
processes.
