What Is The Name Of The Game?

A man wakes up and walks outside of his house. He is then murdered in broad daylight and no one know whom did it. He wakes up in a sort of purgatory in which he meets an androgynous person whom tells him that he will bring him back to life so that he can find his killer and stop himself from being killed. This man travels back and forth through time to find his killer and whenever he does die, he returns to the purgatory place as well as the androgynous person so that he can be revived and finish his quest. I have never completed the game, but I would love to know what it is. The setting is in Japan, I believe. The man has white hair and is very good looking for a game character. The console is XBox (obviously not 360). It is an older game.