Problem:
The stochastic optimal control problem is to minimize the discounted value of an expected weighted average of the squared cash flow $x_t$ and the squared cash flow $y_t$ by choosing the control $i_t$, subject to a lower bound on the constraint: \begin{align} J_t = \inf_{i_t} \ &E_t \int_t^\infty e^{-\delta (u - t)} \left( x_u^2 + \lambda y_u^2 \right) du \\ s.t. \ &i_t \geq - \xi \end{align} where the dynamics of the $x_t$ and $y_t$ are determined jointly by a system of forward-backward stochastic differential equations: \begin{align} dx_t &= \left[ i_t - y_t +b \right] dt + \sigma^x dZ_t \\ \label{eq_a178} dy_t &= \left[ \delta y_t - c x_t \right] dt + \sigma^y dZ_t \end{align}
Question:
Can I ask you to write the Hamiltonian (including the the Karush–Kuhn–Tucker component) and the conditions from the stochastic maximum principle (including the sign of the Karush–Kuhn–Tucker multiplier)?
I have done it and I am happy to provide my attempt. However, I would prefer not to, because I don't want to influence your answer. I just want to verify that my steps are correct.