According to Arsene Wenger's post-game comments in the Guardian, Theo Walcott will remain with Arsenal for the rest of this season:

...asked after Arsenal's 5-2 win over Tottenham – during which Walcott assisted Per Mertesacker's opening goal for the hosts before scoring their final one himself – if the England international would be sold in January if an agreement had not been reached by then, Wenger said he would not. Then, asked if that meant Walcott would be staying until the end of the season, the Frenchman said he would.

-- Source: The Guardian

Walcott, as has been discussed before, wants to play center forward, but with the current form of Olivier Giroud, plus the way that the latter's skillset fits into Arsenal's attack, he will have to be patient. Jack Wilshere wants him to stay anyhow:

"I hope Theo signs [a new contract]...We know what Theo brings; he's got pace, he can go behind, and his movement and timing is probably the best around. He can play anywhere up front but wants to play down the middle, and I think he will get his chance."

This move makes sense for Arsenal; if Walcott were to leave in January, it would be at a cut rate, and keeping him until the summer gives the team six more months of his play as well as more time to convince him to SIGN DA TING. Walcott's been playing very well on the right, and if he keeps it up, hopefully Arsenal can find a way to make it happen.