Can i still join The Dark Brotherhood

While the answer provided by Yasskier is correct in itself, it doesn't focus on your question; can I continue the dark brotherhood quest even though I have found the home of Aventus Aretino without having been given a quest.

The answer is; yes.

After all, the quest to join the Dark Brotherhood is Innocence Lost which only begins when you speak to Aventus - it cannot begin before this point. So, just go on into the house and speak to him.

The "mission" you may be referring to is the miscellaneous objective "Talk to Aventus Aretino" which can be given to you by asking an innkeeper for rumours. This isn't required to begin the main Dark Brotherhood quest, it's just a rumour that points you towards a potential quest. There's a lot of quests that have a rumour as a "pointer", but you can always start them yourself without ever hearing about the rumour; this is the situation you find yourself in.

Rumours usually come from innkeepers or guards and will result in a miscellaneous objective and a quest marker, but other than that can be ignored or followed at your leisure. Other examples include:

• The Black Star - innkeepers will direct you to the Shrine of Azure, or you can find it yourself and start the quest
• Dawnguard (Quest) - guards will point you towards to newly inhabited Fort Dawnguard, or find it yourself
• Surgery - inkeepers and guards will mention the Face Sculptor in Riften, but you can find her yourself

There are quite a few others too.