This paper presents the use of a novel modelling technique based around intermittent transport due to filament motion, to interpret experimental profile and fluctuation data in the scrape-off layer (SOL) of JET during the onset and evolution of a density profile shoulder. Three cases are studied at different points in a density ramp experiment where a shoulder forms in the SOL density profile. A baseline case is established, prior to the shoulder formation, and the stochastic model is shown to be capable of simultaneously matching the time averaged profile measurement as well as the PDF shape and autocorrelation function from the ion-saturation current time series at the outer wall. Aspects of the stochastic model are then varied with the aim of producing a profile shoulder with statistical measurements consistent with experiment. This is achieved through a strong localised reduction in the density sink acting on the filaments within the model. A similar profile shoulder can be achieved by a radial acceleration/deceleration of the filaments, however in this case the autocorrelation function measured at the outer-wall position displays the opposite trend to experiment. In this sense a local reduction in the density sink provides the most consistent manner of inducing a shoulder in the density profile within the stochastic model. The required reduction of the density sink occurs over a highly localised region with the timescale of the density sink increased by a factor of 25. This alone is found to be insufficient to model the expansion and flattening of the shoulder region as the density increases, which requires additional changes within the stochastic model. An example is found which includes both a reduction in the density sink and filament acceleration and provides a consistent match to the experimental data as the shoulder expands, though the uniqueness of this solution can not be guaranteed. Within the context of the stochastic model, this implies that the localised reduction in the density sink can trigger shoulder formation, but additional physics is required to explain the subsequent evolution of the profile. The stochastic model has the potential to be a powerful interpretive tool and suggestions are given towards its improvement and utilisation in the future.