Hi Sukhjinder,
Not sure why the API does not provide that filter, but does provide segmentEnd.
For digital conversations you might need to use the jobs API Conversation Detail job. The "regular" analytics API pulls data of digital conversations that started on the day of in the interval you set, whereas the Jobs API will pull data of conversations that overlap with the interval. E.g. you'd miss a digital conversation in the first one if it was opened again, but the start date was outside the interval you set. Afterwards you'd filter the results on segmentStart in you application.