Robust consumption-investment problems with random market coefficients (Q1938991)

This paper studies a robust consumption-investment problem in a financial market model with general random coefficients and in the presence of uncertainty concerning the true value of the market price of a risk process \(\theta=\{\theta_{t}\}\) which is contained in a compact uncertainty set \({\mathcal U}\). Then, a robust formulation of the consumption-investment problem is given, which is expressed as a max-min problem and it is shown that the value function of this game satisfies a Bellman-Isaacs-type equation (Theorem 1), which allows for a characterization of the optimal policy. The rest of the paper is devoted to a detailed study of this problem for special types of utility functions, and in particular power-, logarithmic- and exponential-type utility functions. For these special cases, it is shown that ansätze for the value function can be employed which involve the starting value of the wealth \(x\) and time \(t\) and a stochastic process \(\{Y_{t}\}\), interpreted as a market state process or information process, which is obtained by the solution of a backward stochastic differential equation. The optimal policy can be characterized in terms of this process.