  Life insurance cash flows become reserve dependent when contract conditions
are modified during the contract term on condition that actuarial equivalence
is maintained. As a result, insurance cash flows and prospective reserves
depend on each other in a circular way, and it is a non-trivial problem to
solve that circularity and make cash flows and prospective reserves
well-defined. In Markovian models, the (stochastic) Thiele equation and the
Cantelli Theorem are the standard tools for solving the circularity issue and
for maintaining actuarial equivalence. This paper expands the stochastic Thiele
equation and the Cantelli Theorem to non-Markovian frameworks and presents a
recursive scheme for the calculation of multiple contract modifications.
