  We investigate in this work a recently proposed diagrammatic quantum Monte
Carlo method --- the inchworm Monte Carlo method --- for open quantum systems.
We establish its validity rigorously based on resummation of Dyson series.
Moreover, we introduce an integro-differential equation formulation for open
quantum systems, which illuminates the mathematical structure of the inchworm
algorithm. This new formulation leads to an improvement of the inchworm
algorithm by introducing classical deterministic time-integration schemes. The
numerical method is validated by applications to the spin-boson model.
