Stochastic integration with respect to q Brownian motion

We develop a stochastic integration with respect to a q-Brownian motion (for -1 < q < 1), i.e. a non commutative process X(t) = a(t) + a^*(t)where the operator a(t) and its adjoint fulfill the q commutation relation a(s) a^*(t) - q a^*(t) a(s) = (s \wedge t) 1; under the vacuum state expectation. We show that this process enjoys a predictable representation type property.