  Higher-order optimization problems naturally appear when investigating the
effects of a patent with finite length, as in the pioneering work of Futagami
and Iwaisako (2007). In this paper, we establish the Euler equations and
transversality conditions necessary for analyzing such higher-order
optimization problems. We develop our results for stochastic general
reduced-form models and consider cases of both continuous and discrete time. We
employ our results to establish the Euler equations and transversality
conditions for the simplified household maximization problem in Futagami and
Iwaisako (2007).
