Both nations have a claim to Kashmir. Pakistan cites the terms of Partition of the subcontinent, when areas of Muslim majority went to Pakistan. However, the terms applied only to British territory and not to Kashmir, which was ruled by Maharaja Hari Singh. Both new born countries lobbied hard for his allegiance, but he aligned with India, for his own interests.

I quote Prof. Sumantra Bose, a well-respected researcher of the Kashmir problem.

Ultimately, the Indians gained Kashmir because the most important political movement of that time, the National Conference Movement of the Kashmiri Muslims preferred India to Pakistan, if it came to a choice between these two. So it was not just the Maharaja's signature that sealed the deal for India. National Conference was led by a very charismatic Kashmiri leader called Sheikh Mohammad Abdullah. Sheikh Abdullah was known at that time as Sher-e-Kashmir, the Lion of Kashmir, and there was little doubt about the fact that he preferred India to Pakistan. This preference was a decisive factor.

Why are Pakistanis reluctant to acknowledge the importance of this factor? It is because it blows a big hole in the myth of Muslim solidarity. It shows that there were influential Muslim groups, including Sheikh Abdullah's movement in Jammu and Kashmir, which were not in favor of Pakistan simply because they were Muslims. They were prepared to take their chances in India and test India's claims to being a secular, tolerant and inclusive republic.

-- Prof. Sumantra Bose in a 2001 interview with Fathom.

I personally think Kashmir belonged to India than to Pakistan. Kashmir in 1947 had about 30% Hindus, and they were unwelcome in Pakistan as it was an Islamic nation. But both Hindus and Muslims could live in India due to its secular constitution.

Pakistan feels that the referendum be held in Indian part of Kashmir asking the people whether they'd like to affiliate with India or Pakistan. Some in India feel that the referendum should include whole of Kashmir, including the Pakistan occupied Kashmir, and a third option, for creation of a nation of Kashmir be provided on the ballot.

While people determining to which nation they should belong seems like a legitimate solution, it certainly is not a fair one. Imagine what would have happened if Abraham Lincoln went in for a referendum in the South instead of the war -- the South most certainly would have seceded. Another instance is Bosnia, where a referendum on independence, which was massively boycotted by the Bosnian Serbs who made up about 35% of the population in 1992, made civil war inevitable.

India will never agree to a referendum because numerous other states within India over the years (notably Punjab and Assam) have asked to be independent nations. It would be a bad (and IMHO unfair) precedent.  Indians believe that as long as elections are held for local governments in Kashmir, they are honoring people's will.

There was peace in Kashmir till 1989. 

The solution is co-habitation of Hindus and Muslims as they have done so for centuries. Pakistan must realize that it will never, ever will be able to control Indian administered Kashmir. Muslims outside of India must realize that Kashmir is not a Hindu-Muslim problem, because there are Muslims living in each and every village of India, who will never be able to settle in Pakistan. The Indian Government must do a better job educating the people of Pakistan that they are supporting an unjust cause.