From what I understand about the big bang theory, everything was compressed in a tiny space before it happened, which would theoretically result in a black hole. Now, it is theorized that black holes can explode in on themselves, which could, in theory, explain the big bang. As far as I know though, we have never actually witnessed a black hole exploding in on itself. We have seen black holes cause explosions, but the black hole continues to exist and the method of creating the explosion is different. And the idea of black holes being able to explode in on themselves is based in quantum physics, which, I personally don't buy into, and many scientists can point out problems with it. Quantum mechanics is based on treating probabilities as if they are all occurring at once (until they are observed). I'd recommend looking up Schrodinger's cat. Keep in mind, Schrodinger created the thought experiment to show that this model of looking at things was dumb, because us observing a system does not change anything in that system. Quantum mechanics is kind of like asking if the wall is there when we're not looking. I much prefer Einstein's theory of general relativity over quantum theory.
In short, it's possible in quantum theory, but I really think that theory is at least premised on bogus illogic, and the only other explanation I see is a divine creator.