Transversality Conditions for Stochastic Higher-Order Optimality: Continuous and Discrete Time Problems

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).