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 $i_t$, which is subject to a lower bound 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} The dynamics of the states $x_t$ 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}
My approach:
The Hamiltonian is given by: \begin{align} H_t = x_t^2 + b_t^2 + E_t \left[ \frac{dJ_t}{dt} \right] - \mu_i (i_t + \xi) \end{align} where: \begin{align} E_t \left[ \frac{dJ_t}{dt} \right] &= J_{x_t} E_t \left[ \frac{dx_t}{dt} \right] + \frac{1}{2} J_{x_t x_t} E_t \left[ \frac{(dx_t)^2}{dt} \right] + J_{y_t} E_t \left[ \frac{dy_t}{dt} \right] + \frac{1}{2} J_{y_t y_t} E_t \left[ \frac{(dy_t)^2}{dt} \right] + J_{x_t y_t} E_t \left[ \frac{dx_t dy_t}{dt} \right] \nonumber \\ &= J_{x_t} \left[ i_t - y_t + b \right] + \frac{1}{2} J_{x_t x_t} (\sigma^x)^2 + J_{y_t} \left[ \delta y_t - c x_t \right] + \frac{1}{2} J_{y_t y_t} (\sigma_t^y)^2 + J_{x_t y_t} \rho_t^{x y} \sigma^x \sigma_t^y \nonumber \end{align} Next, I find the optimality conditions: \begin{align} 0 &= H_{i_t} \\ J_{x_t} &= \mu_i \nonumber \end{align} \begin{align} 0 &= H_{x_t} + E_t \left[ \frac{dJ_{x_t}}{dt} \right] - \delta J_{x_t} \\ 0 &= 2 x_t - J_{y_t} c + E_t \left[ \frac{dJ_{x_t}}{dt} \right] - \delta J_{x_t} \nonumber \end{align} \begin{align} 0 &= H_{y_t} + E_t \left[ \frac{dJ_{y_t}}{dt} \right] - \delta J_{y_t} \\ 0 &= 2 \lambda y_t - J_{x_t} + \delta J_{y_t} + E_t \left[ \frac{dJ_{y_t}}{dt} \right] - \delta J_{y_t} \nonumber \\ 0 &= 2 \lambda y_t - J_{x_t} + E_t \left[ \frac{dJ_{y_t}}{dt} \right] \nonumber \end{align} \begin{align} 0 &= \mu_i (i_t + \xi) \\ 0 &= J_{x_t} (i_t + \xi) \nonumber \end{align} \begin{align} 0 \leq \mu_i \\ 0 \leq J_{x_t} \nonumber \end{align}
Question:
Can you write the Hamiltonian (including the the Karush–Kuhn–Tucker component) and the conditions from the stochastic minimum principle so I can verify whether my approach is correct?
In particular, I have the following doubt: is my Hamiltonian correct, or should it be: \begin{align} H_t = - x_t^2 - b_t^2 + E_t \left[ \frac{dJ_t}{dt} \right] + \mu_i (i_t + \xi) \end{align} This in turn would flip the sign of $J_{x_t}$.