Reasoning about Recursive Probabilistic Programs
  This paper presents a wp-style calculus for obtaining expectations on the outcomes of (mutually) recursive probabilistic programs. We provide several proof rules to derive one-- and two--sided bounds for such expectations, and show the soundness of our wp-calculus with respect to a probabilistic pushdown automaton semantics. We also give a wp-style calculus for obtaining bounds on the expected runtime of recursive programs that can be used to determine the (possibly infinite) time until termination of such programs. 
Federico Olmedo and Benjamin Lucien Kaminski and Joost-Pieter Katoen   and Christoph Matheja
cs.LO