  We present a set of tools to assess the capabilities of LISA to detect and
reconstruct the spectral shape and amplitude of a stochastic gravitational wave
background (SGWB). We first provide the LISA power-law sensitivity curve and
binned power-law sensitivity curves, based on the latest updates on the LISA
design. These curves are useful to make a qualitative assessment of the
detection and reconstruction prospects of a SGWB. For a quantitative
reconstruction of a SGWB with arbitrary power spectrum shape, we propose a
novel data analysis technique: by means of an automatized adaptive procedure,
we conveniently split the LISA sensitivity band into frequency bins, and fit
the data inside each bin with a power law signal plus a model of the
instrumental noise. We apply the procedure to SGWB signals with a variety of
representative frequency profiles, and prove that LISA can reconstruct their
spectral shape. Our procedure, implemented in the code SGWBinner, is suitable
for homogeneous and isotropic SGWBs detectable at LISA, and it is also expected
to work for other gravitational wave observatories.
