  Finding a stochastic gravitational-wave background (SGWB) of astrophysical or
primordial origin is one of the quests of current and future gravitational-wave
observatories. While detector networks such as LIGO-Virgo-Kagra or pulsar
timing arrays can use cross-correlations to tell instrumental noise and SGWB
apart, LISA is likely to be the only flying detector of its kind in 2035. This
particularity poses a challenge for data analysis. To tackle it, we present a
strategy based on Bayesian model selection. We use a flexible noise power
spectral density~(PSD) model and the knowledge of noise and signal transfer
functions to allow SGWBs detection when the noise PSD is unknown. With this
technique, we then probe the parameter space accessible by LISA for power-law
SGWB shapes.
