Private-equity firms and hedge funds are increasing their control of the rights to collect America’s monthly mortgage payments, an almost $10 trillion market that banks are retreating from amid looming regulations.

Servicing rights, or MSRs, on at least $1 trillion of mortgages will trade in the next two years, said Jay Bray, chief executive officer of Nationstar Mortgage Holdings Inc., a servicer majority owned by Fortress Investment Group LLC. The private equity firm said in July it raised a new $1.1 billion fund to buy the contracts. GSO Capital Partners LP, the credit arm of Blackstone Group LP, announced last week it helped form a company targeting the assets after Two Harbors Investment Corp., a real estate investment trust, started buying the contracts in the second quarter.