Sincov's and other functional equations and negative interest rates

Investigating the future value $F(K,s,t)$ of a capital $K$ invested at date $S$ at date $t$ the “natural” condition $F(K,s,t)\ge K$ has lost its naturality because of the strange fact of negative interest rates. This leads to the task of describing the possible solutions of the multiplicative Sincov equation $f(s,u)=f(s,t)f(t,u)$ for $s\le t\le u$, where $f(s,t)=0$ may happen. In this paper we solve this task and discuss connections to the theory of investments.