We study the minimization of the expected costs under stochastic constraint at the terminal time. The first and the main result says that for a power type of costs, the value function is the minimal positive solution of a second order semi-linear ordinary differential equation (ODE). Moreover, we establish the optimal control. In the second example we show that the case of exponential costs leads to a trivial optimal control.
Yan Dolinsky. Benjamin Gottesman. Gurel-Gurevich Ori. "A note on costs minimization with stochastic target constraints." Electron. Commun. Probab. 25 1 - 12, 2020. https://doi.org/10.1214/20-ECP295