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Article

HJB equations with gradient constraint associated with jump-diffusion controlled processes

Moreno-Franco H. A., Kelbert M.

We guarantee the existence and uniqueness (in the almost everywhere sense) of the solution to a Hamilton-Jacobi-Bellman (HJB) equation with gradient constraint and an integro-differential operator whose Lévy measure has bounded variation. This type of equation arises in singular stochastic control problems where the state process is a jump-diffusion with infinite activity and finite variation jumps. By means of ε-penalized controls we show that the value function associated to this class of problems agrees with the solution to our HJB equation.