Stochastic impulse control Problem with state and time dependent cost functions

Abstract

<p style='text-indent:20px;'>We consider stochastic impulse control problems when the impulses cost functions depend on <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ x $\end{document}</tex-math></inline-formula>. We use the approximation scheme and viscosity solutions approach to show that the value function is a unique viscosity solution for the associated Hamilton-Jacobi-Bellman equation (HJB) partial differential equation (PDE) of stochastic impulse control problems.