I have the following stochastic optimal control problem, where
$$dq_t=u_tdt+\sigma (1+z_t) W_t$$
and $q_0 \in \mathbb{R}$, $\sigma>0$ and where $W_t$ is the Wiener process/Brownian motion, and the cost functional is,
$$J(u,z) = \mathbb{E}\left[\int_0^T \left(q_t (s_0 -aq_t) - \frac{\kappa_t}{2}u_t^2 - \frac{g_t}{2}z_t^2 \right)dt \right] + \mathbb{E}\left[\lambda a \left(\frac{d\mathbb{Q}}{d\mathbb{P}}-1\right)\left(\frac{s_0}{a}-q_T\right)^2 \right],$$
where $(\kappa_t)_{0 \leq t \leq T}$ and $(g _t)_{0 \leq t \leq T}$ are progressively measurable strict positive processes, $s_o, a, \kappa, g, T > 0$ and $\lambda \in \mathbb{R}$. In addition $q$ is a martingale under the measure $\mathbb{Q}$ and $\frac{d\mathbb{Q}}{d\mathbb{P}}$ is the associated density process and is given by $$\frac{d\mathbb{Q}}{d\mathbb{P}} = \exp{\left\{-\int_0^T \delta_t dW_t - \frac{1}{2}\int_0^T \delta_t^2 dt \right\}}, \quad \delta_t = \frac{u_t}{\sigma (1+ z_t)}.$$
So I got the following optimization problem, $$\sup_{u,z} J (u,z).$$
Now I would like to solve this problem and provide an optimizer in feedback form. Unfortunately, I can't solve it. I tried to use the results from [1] and [2], but they require that $\lambda a \left(\frac{d\mathbb{Q}}{d\mathbb{P}}-1\right)$ has to be positive (in my problem it can also be negative).
Is there any result that involves my case? Can someone help me here or give me tips? I really appreciate any help you can provide.
PS:I know, that the only possibility is using the Pontryagin maximum principle and backward stochastic differential equations to solve the problem.