2 research outputs found
Optimal Consumption--Investment Problems under Time-Varying Incomplete Preferences
The main objective of this paper is to develop a martingale-type solution to
optimal consumption--investment choice problems ([Merton, 1969] and [Merton,
1971]) under time-varying incomplete preferences driven by externalities such
as patience, socialization effects, and market volatility. The market is
composed of multiple risky assets and multiple consumption goods, while in
addition there are multiple fluctuating preference parameters with inexact
values connected to imprecise tastes. Utility maximization is a multi-criteria
problem with possibly function-valued criteria. To come up with a complete
characterization of the solutions, first we motivate and introduce a set-valued
stochastic process for the dynamics of multi-utility indices and formulate the
optimization problem in a topological vector space. Then, we modify a classical
scalarization method allowing for infiniteness and randomness in dimensions and
prove results of equivalence to the original problem. Illustrative examples are
given to demonstrate practical interests and method applicability
progressively. The link between the original problem and a dual problem is also
discussed, relatively briefly. Finally, using Malliavin calculus with
stochastic geometry, we find optimal investment policies to be generally
set-valued, each of whose selectors admits a four-way decomposition involving
an additional indecisiveness risk-hedging portfolio. Our results touch on new
directions for optimal consumption--investment choices in the presence of
incomparability and time inconsistency, also signaling potentially testable
assumptions on the variability of asset prices. Simulation techniques for
set-valued processes are studied for how solved optimal policies can be
computed in practice.Comment: 72 pages, 1 table, 13 figure