I am looking for references that explain how to solve stochastic optimal control problems in continuous time and where the control variable is subject to a lower bound constraint. I have been unable to find books or papers that show how to approach this problem. Could anyone help?
More precisely, 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} I have fully solved the problem, finding the second state $y_t$, control $i_t$, and co-states $J_{x_t}$ and $J_{\pi_t}$ as functions of the state $x_t$. To do so, I have solved the problem in two regions ($- \infty < x_t < \hat{x}$ where $i_t = - \xi$, and $\hat{x} < x_t < \infty$ where $i_t > - \xi$), where $\hat{x}$ denotes the value of $x_t$ at which the control jumps to the lower bound. The results is that the control $i_t$ is a linear function of $x_t$ for $x_t > \hat{x}$, while it is equal to the lower bound $i_t = -\xi$ when $x_t < \hat{x}$. However, I cannot figure out the value of $\hat{x}$. One may be tempted to say that $\hat{x}$ is the value of $x_t$ at which $i(x_t) = -\xi$, where $i(x_t)$ is the linear function I just talked about. But this violates the positivity constraint of the Lagrange multiplier. Therefore, it must be that $i(x_t)$ transitions to $i_t = -\xi$ with a jump.