Cadlag Skorokhod problem driven by a maximal monotone operator

Abstract

The article deals with existence and uniqueness of the solution of the following differential equation (a c\`adl\`ag Skorokhod problem) driven by a maximal monotone operator and with singular input generated by the c\`{a}dl\`{a}g function $m$: \[ \left\{ \begin{array} [c]{l} dx_{t}+A\left( x_{t}\right) \left( dt\right) +dk_{t}^{d}\ni dm_{t} \,,~t\geq0,\\ x_{0}=m_{0}, \end{array} \right. \] where $k^{d}$ is a pure jump function. The jumps outside of the constrained domain $\overline{\mathrm{D}(A)}$ are counteracted through the generalized projection $\Pi$, by taking $x_{t}=\Pi(x_{t-}+\Delta m_{t})$, whenever $x_{t-}+\Delta m_{t}\notin\overline {\mathrm{D}(A)}\,$. Approximations of the solution based on discretization and Yosida penalization are considered.