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This paper is concerned with the axiomatic foundation and explicitconstruction of a general class of optimality criteria that can be used forinvestment problems with multiple time horizons, or when the time horizon isnot known in advance. Both the investment criterion and the optimal strategyare characterized by the Hamilton-Jacobi-Bellman equation on a semi-infinitetime interval. In the case when this equation can be linearized, the problemreduces to a time-reversed parabolic equation, which cannot be analyzed via thestandard methods of partial differential equations. Under the additionaluniform ellipticity condition, we make use of the available description of allminimal solutions to such equations, along with some basic facts from potentialtheory and convex analysis, to obtain an explicit integral representation ofall positive solutions. These results allow us to construct a large family ofthe aforementioned optimality criteria, including some closed form examples inrelevant financial models.

Optimal investment for all time horizons and Martin boundary of space-time diffusions