  The detection of a stochastic background of gravitational waves could
significantly impact our understanding of the physical processes that shaped
the early Universe. The challenge lies in separating the cosmological signal
from other stochastic processes such as instrument noise and astrophysical
foregrounds. One approach is to build two or more detectors and cross correlate
their output, thereby enhancing the common gravitational wave signal relative
to the uncorrelated instrument noise. When only one detector is available, as
will likely be the case with the Laser Interferometer Space Antenna (LISA),
alternative analysis techniques must be developed. Here we show that models of
the noise and signal transfer functions can be used to tease apart the
gravitational and instrument noise contributions. We discuss the role of
gravitational wave insensitive "null channels" formed from particular
combinations of the time delay interferometry, and derive a new combination
that maintains this insensitivity for unequal arm length detectors. We show
that, in the absence of astrophysical foregrounds, LISA could detect signals
with energy densities as low as $\Omega_{\rm gw} = 6 \times 10^{-13}$ with just
one month of data. We describe an end-to-end Bayesian analysis pipeline that is
able to search for, characterize and assign confidence levels for the detection
of a stochastic gravitational wave background, and demonstrate the
effectiveness of this approach using simulated data from the third round of
Mock LISA Data Challenges.
