From India perspective don't think there is any restriction. But each country has its own rules and regulation about how and what a foreign resident can trade. Transferring of funds will have to be as per RBI guidelines.
But from taxation perspective it becomes a bit complicated. Any income will have to be reported while filling IT returns in India. Also there are local taxation laws that could apply. India does have double taxation treaties with some countries which could help.
Check out following for how to trade in USA
http://tradingqna.com/3026/can-we-trade-in-u-s-stock-market-using-zerodha