Dust particles observed in extrasolar planetary discs originate from undetectable km-sized bodies but this valuable information remains uninteresting if the theoretical link between grains and planetesimals is not properly known .
We outline in this paper a numerical approach we developed in order to address this issue for the case of dust producing collisional cascades .
The model is based on a particle-in-a-box method .
We follow the size distribution of particles over eight orders of magnitude in radius taking into account fragmentation and cratering according to different prescriptions .
A very particular attention is paid to the smallest particles , close to the radiation pressure induced cut-off size R _ { pr } , which are placed on highly eccentric orbits by the stellar radiation pressure .
We applied our model to the case of the inner ( < 10 AU ) $ β $ Pictoris disc , in order to quantitatively derive the population of progenitors needed to produce the small amount of dust observed in this region ( \simeq 10 ^ { 22 } g ) .
Our simulations show that the collisional cascade from kilometre-sized bodies to grains significantly departs from the classical dN \propto R ^ { -3.5 } dR power law : the smallest particles ( R \simeq R _ { pr } ) are strongly depleted while an overabundance of grains with size \sim 2 R _ { pr } and a drop of grains with size \sim 100 R _ { pr } develop regardless of disc ’ s dynamical excitation , R _ { pr } 
and initial surface density .
However , the global dust to planetesimal mass ratio remains close to its dN \propto R ^ { -3.5 } dR 
value .
Our rigorous approach thus confirms the depletion in mass in the inner $ β $ Pictoris disc initially inferred from questionable assumptions .
We show moreover that collisions are a sufficient source of dust in the inner $ β $ Pictoris disc .
They are actually unavoidable even when considering the alternative scenario of dust production by slow evaporation of km-sized bodies .
We obtain an upper limit of \sim 0.1 M _ { \oplus } for the total disc mass below 10 AU .
This upper limit is not consistent with the independent mass estimate ( at least 15 M _ { \oplus } ) in the frame of the Falling Evaporating Bodies ( FEB ) scenario explaining the observed transient features activity .
Furthermore , we show that the mass required to sustain the FEB activity implies a so important mass loss that the phenomena should naturally end in less than 1 Myr , namely in less than one twentieth the age of the star ( at least 2 10 ^ { 7 } years ) .
In conclusion , these results might help converge towards a coherent picture of the inner $ β $ Pictoris system : a low-mass disc of collisional debris leftover after the possible formation of planetary embryos , a result which would be coherent with the estimated age of the system .