Before revision:  The strategy we adopt for the existence part is based on approximations using stochastic Volterra equations with L^2-kernels  . Most importantly, we establish weak uniqueness using a duality argument on the Fourier-Laplace transform via a deterministic Riccati-Volterra integral equation.
After revision:  The strategy we adopt for the existence part is based on approximations using stochastic Volterra equations with L^2-kernels combined with a general stability result . Most importantly, we establish weak uniqueness using a duality argument on the Fourier-Laplace transform via a deterministic Riccati-Volterra integral equation.