God exists. In Hinduism God is beyond definition, beyond the mind, and more. There is no single understanding, yet all understandings are correct in their own way, even yours.
Has it occurred to you that YOU ARE GOD? It's like you wake up one day, and you realize that you are god. Not in the sense, though, that "Oh, I'm Allah resurrected! Everyone shall worship me at the Kaaba henceforth." Rather, in the sense that "Oh, what I thought was god is actually me, with the distorted, mirrored reflection of myself removed so that I can see myself. I am a part of it, and it is a part of me, and yet, that is not an accurate way to say this."
Just because we don't have a theoloy of god, doesn't mean we are believing a fake identity. It's reasonable for us to not know all the answers about god for we are just a bit of his mighty creation.