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In singular stochastic control, a control is a finite-variation process rather than only a rate integrated against time. Its increments can have atoms or a part singular with respect to time. The Hamilton-Jacobi-Bellman equation becomes a variational inequality with a constraint on directional derivatives of the value function.

A stochastic control problem chooses an admissible control input using information available in a filtration to optimize an expected objective for a random dynamical system. Dynamic programming often characterizes its value function through a Hamilton-Jacobi-Bellman equation.