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Lets say I am playing against a good friend of mine and I lend him one of my goblin deck.

He then cast Increasing Vengeance from  his graveyard using its flashback ability to copy whatever.

I am play a red/blue spell deck.

Can I cast Innistrad's Runic Repetition targeting the Vengeance since I tecnically "own" the card or the "ownership" of the card depend on who starting playing with their respective deck?

I guess this is not a legal move but I haven't found any rules about the "ownership" of cards to confirm it. 

Thank You. 

for the purpose of the game your friend is the owner of "your" deck
that you only loaned it to him does not matter

you cannot cast Runic Repetition on "his" card

"Own" means "began the game with this in your deck" in magic. It has nothing to do with legal ownership.

If you want the specific rule it's: 
108.3. The owner of a card in the game is the player who started the game with it in his or her deck. If a card is brought into the game from outside the game rather than starting in a player's deck, its owner is the player who brought it into the game. If a card starts the game in the command zone, its owner is the player who put it into the command zone to start the game. Legal ownership of a card in the game is irrelevant to the game rules except for the rules for ante. (See rule 407.)

That's what I thought. Thanks guys.