The Stochastic Turnpike Property without Uniformity in Convex Aggregate Growth Models

An important stochastic turnpike property in optimal growth models asserts that optimal programs of capital accumulation from different initial stocks converge almost surely in a suitable metric. Its proof requires constructing a value-loss process satisfying both uniform boundedness in expectation and sensitivity (in the sense of recording a strictly positive value-loss when the capital stocks being compared diverge). Uniformity assumptions strengthen sensitivity by ensuring that value-loss is independent of time and state of environment in which the divergence occurs. They are imposed either directly on the value-loss process, or indirectly through bounds on the degree of concavity of the felicity or production functions, and are acknowledged as strong restrictions on the model. This paper argues, within the context of a convex aggregate growth model, that uncertainty can obviate the need for uniformity. The multiplicity of states afforded by a stochastic framework permits constructing a value-loss process over an "extended" time-line that is a martingale and, hence, relatively easy to uniformly bound in expectation. Further, if capital stocks diverge by some critical amount in any time and state, then the martingale registers an upcrossing across a band of uniform width on its extended time-line for that state thereby giving uniform value-loss. Probabilistic arguments based on the Martingale Upcrossing theorem and the Borel-Cantelli lemma then clinch the turnpike property.