Database
[{_id:"this won't match",subEntities:[{subId:"999",dateOfStart:"2098-01-01",dateOfTermination:"2099-12-31"},{subId:"998",dateOfStart:"2088-01-01",dateOfTermination:"2089-12-31"}]},{_id:"this will match",subEntities:[{subId:"999",dateOfStart:"2098-01-01",dateOfTermination:"2099-12-31"},{subId:"998",dateOfStart:"2088-01-01",dateOfTermination:"2099-12-31"}]}]
Query
db.collection.find({subEntities:{$elemMatch:{dateOfStart:{"$lte":"2098-01-02"},dateOfTermination:{"$gte":"2099-12-30"},subId:{"$in":["998"]}}}})