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The main objective of this paper is to develop a martingale-type solution tooptimal consumption--investment choice problems ([Merton, 1969] and [Merton,1971]) under time-varying incomplete preferences driven by externalities suchas patience, socialization effects, and market volatility. The market iscomposed of multiple risky assets and multiple consumption goods, while inaddition there are multiple fluctuating preference parameters with inexactvalues connected to imprecise tastes. Utility maximization is a multi-criteriaproblem with possibly function-valued criteria. To come up with a completecharacterization of the solutions, first we motivate and introduce a set-valuedstochastic process for the dynamics of multi-utility indices and formulate theoptimization problem in a topological vector space. Then, we modify a classicalscalarization method allowing for infiniteness and randomness in dimensions andprove results of equivalence to the original problem. Illustrative examples aregiven to demonstrate practical interests and method applicabilityprogressively. The link between the original problem and a dual problem is alsodiscussed, relatively briefly. Finally, using Malliavin calculus withstochastic geometry, we find optimal investment policies to be generallyset-valued, each of whose selectors admits a four-way decomposition involvingan additional indecisiveness risk-hedging portfolio. Our results touch on newdirections for optimal consumption--investment choices in the presence ofincomparability and time inconsistency, also signaling potentially testableassumptions on the variability of asset prices. Simulation techniques forset-valued processes are studied for how solved optimal policies can becomputed in practice.

Optimal Consumption--Investment Problems under Time-Varying Incomplete Preferences