ASRS 2084272

We were given very tight vectors from the west to join the LOC-DME/E at ASE, with a strong wind from the west, causing us to overshoot final. The autopilot was correcting as were were descending, but I picked up and verified ASE runway and decided to proceed visually as we were VMC and clear of clouds. I continued to correct back to final but the wind made that very challenging. I proceeded visually to descend and line up on final. In the heat of the moment, I missed whether we were updated to cleared visual approach, but continued to descend and line up on final due to the critical nature of descending to prevent a very steep approach at ASE. We were VMC the whole way and correcting to final. The approach was controlled and very normal for an approach into ASE. Once we landed, we were asked to call tower to discuss the event. They relayed to us that they had had a rash of overshooting finals lately and that pilots were descending below published altitudes. For us, the problem started with the tight vectors to the overshoot with the west wind complicating getting back to final. Once below the clouds, I descended to prevent an overly steep approach, which is what you get if you fly it as published. In the heat of the moment correcting back to center and descending, I missed whether we had been updated from cleared the LOC-DME/E to a visual approach. In the future, I will make sure we have been cleared to proceed visually before descending to prevent the steep approach.