# A Note on Costs Minimization with Stochastic Target Constraints

## TL;DR

This paper investigates minimizing expected costs under stochastic terminal constraints, deriving a differential equation for power costs and analyzing the triviality of exponential costs in optimal control.

## Contribution

It provides a novel characterization of the value function via a semi-linear ODE for power costs and explores the case of exponential costs.

## Key findings

- Value function is the minimal positive solution of a semi-linear ODE for power costs.
- Exponential costs lead to a trivial optimal control.
- Established the form of the optimal control under stochastic terminal constraints.

## Abstract

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.