Cadlag Skorokhod problem driven by a maximal monotone operator

TL;DR

This paper establishes existence and uniqueness of solutions for a Skorokhod problem driven by a maximal monotone operator with singular, cadlag input, and introduces approximation methods including discretization and Yosida penalization.

Contribution

It provides new results on the well-posedness of a Skorokhod problem with maximal monotone operators and singular inputs, along with approximation techniques.

Findings

Existence and uniqueness of solutions are proven.

Approximation schemes via discretization are developed.

Yosida penalization effectively approximates solutions.

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.