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We use the latest constraints on the population of stellar origin binaryblack holes (SOBBH) from LIGO/Virgo/KAGRA (LVK) observations, to estimate thestochastic gravitational wave background (SGWB) they generate in the frequencyband of LISA. We account for the faint and distant binaries, which contributethe most to the SGWB, by extending the merger rate at high redshift assuming ittracks the star formation rate. We adopt different methods to compute the SGWBsignal: an analytical evaluation, Monte Carlo sums over SOBBH populationrealisations, and a method that accounts for the role of the detector bysimulating LISA data and iteratively removing resolvable signals until only theconfusion noise is left, allowing for the extraction of both the expected SGWBand the number of resolvable SOBBHs. Since the latter are few for SNRthresholds larger than five, we confirm that the spectral shape of the SGWB inthe LISA band follows the analytical prediction of a power law. We infer theprobability distribution of the SGWB amplitude from the LVK GWTC-3 posterior ofthe 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. Weperform a MC analysis to assess LISA's capability to detect and characterisethis signal. Accounting for both the instrumental noise and the galacticbinaries foreground, with four years of data, LISA will be able to detect theSOBBH SGWB with percent accuracy, narrowing down the uncertainty on theamplitude by one order of magnitude with respect to the range of possibleamplitudes inferred from the population model. A measurement of this signal byLISA will help to break the degeneracy among some of the population parameters,and provide interesting constraints, in particular on the redshift evolution ofthe SOBBH merger rate.

Stochastic gravitational wave background from stellar origin binary black holes in LISA