Classical stochastic optimal control problem is to minimize functional $$ J(u) = \mathbf E \int_0^T f(t,x_t,u_t)dt, \tag{1} $$ subject to SDE $$ dx_t = b(t,x_t,u_t)dt + \sigma(t,x_t,u_t)dW_t, \quad x_0 = \mathbf x, \tag{2} $$ where $b$ and $\sigma$ are random functions: $\mathbf R_+\times\mathbf R\times\mathbf R\times\Omega \to \mathbf R$. As far as I know there are two major approaches to such problems: maximum principle and dynamic programming.

My question is how to solve optimal control problems when instead of equation (2) we have the following equation$$ dx_t = b(t,x_t,u_t)dt, \quad x_0 = \mathbf x, \tag{3} $$ (same random drift coefficient $b$, but no diffusion $\sigma$)?. Is there special theory for this case or can we apply same methods as problem (1)-(2)?

update: To make it clear, drift is of the following form $$b=b(t,x_t,u_t,\omega):\mathbf R_+\times\mathbf R\times\mathbf R\times\Omega \to \mathbf R,$$ as particular case $b=b(t,x_t,u_t,\xi_t)$, where $\xi_t$ is some other process.