This effect has no duration. The targeted player retains control of the targeted permanent until the game ends, the permanent leaves the battlefield, or an effect causes someone else to gain control of it.

If either the targeted player or the targeted permanent is an illegal target by the time the ability resolves, the ability does nothing. The player won’t gain control of the permanent.

You may target yourself with Bazaar Trader’s ability. Normally, this won’t have any visible effect. However, the ability would override an effect with a limited duration that gave you control of a permanent. For example, if you temporarily gained control of a creature with Act of Treason, targeting yourself and that creature with Bazaar Trader’s ability would then cause you to gain control of the creature indefinitely.