Reflecting thoughts

The purpose of this note is to provide an equivalent definition and an alternative proof of uniqueness of the one-dimensional reflection map which is more a direct derivation that structurally leads to the form of the map when it exists, does not involve integration (neither in the definition nor in the proof) and for which no assumptions on the driving process is needed. Also, it is argued that, with the proposed definition, the reflection map exists provided that the driving process is lower semicontinuous from the right, but is not necessarily right continuous and does not necessarily have left limits. These ideas are then easily extended to the multidimensional case.