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We use the latest constraints on the population of stellar origin binary
black holes (SOBBH) from LIGO/Virgo/KAGRA (LVK) observations, to estimate the
stochastic gravitational wave background (SGWB) they generate in the frequency
band of LISA. We account for the faint and distant binaries, which contribute
the most to the SGWB, by extending the merger rate at high redshift assuming it
tracks the star formation rate. We adopt different methods to compute the SGWB
signal: an analytical evaluation, Monte Carlo sums over SOBBH population
realisations, and a method that accounts for the role of the detector by
simulating LISA data and iteratively removing resolvable signals until only the
confusion noise is left, allowing for the extraction of both the expected SGWB
and the number of resolvable SOBBHs. Since the latter are few for SNR
thresholds larger than five, we confirm that the spectral shape of the SGWB in
the LISA band follows the analytical prediction of a power law. We infer the
probability distribution of the SGWB amplitude from the LVK GWTC-3 posterior of
the binary population model; its interquartile range of
$h^2\Omega_\mathrm{GW}(f=3\times10^{-3}\,\mathrm{Hz}) \in
[5.65,\,11.5]\times10^{-13}$ is in agreement with most previous estimates. We
perform a MC analysis to assess LISA's capability to detect and characterise
this signal. Accounting for both the instrumental noise and the galactic
binaries foreground, with four years of data, LISA will be able to detect the
SOBBH SGWB with percent accuracy, narrowing down the uncertainty on the
amplitude by one order of magnitude with respect to the range of possible
amplitudes inferred from the population model. A measurement of this signal by
LISA will help to break the degeneracy among some of the population parameters,
and provide interesting constraints, in particular on the redshift evolution of
the SOBBH merger rate.