  We propose and analyze the convergence of a novel stochastic forward-backward
splitting algorithm for solving monotone inclusions given by the sum of a
maximal monotone operator and a single-valued maximal monotone cocoercive
operator. This latter framework has a number of interesting special cases,
including variational inequalities and convex minimization problems, while
stochastic approaches are practically relevant to account for perturbations in
the data. The algorithm we propose is a stochastic extension of the classical
deterministic forward-backward method, and is obtained considering the
composition of the resolvent of the maximal monotone operator with a forward
step based on a stochastic estimate of the single-valued operator. Our study
provides a non-asymptotic error analysis in expectation for the strongly
monotone case, as well as almost sure convergence under weaker assumptions. The
approach we consider allows to avoid averaging, a feature critical when
considering methods based on sparsity, and, for minimization problems, it
allows to obtain convergence rates matching those obtained by stochastic
extensions of so called accelerated methods. Stochastic quasi Fejer's sequences
are a key technical tool to prove almost sure convergence.
