Abstract
In this paper we investigate a utility maximization problem with drift uncertainty in a multivariate continuous-time Black–Scholes type financial market which may be incomplete. We impose a constraint on the admissible strategies that prevents a pure bond investment and we include uncertainty by means of ellipsoidal uncertainty sets for the drift. Our main results consist firstly in finding an explicit representation of the optimal strategy and the worst-case parameter, secondly in proving a minimax theorem that connects our robust utility maximization problem with the corresponding dual problem. Thirdly, we show that, as the degree of model uncertainty increases, the optimal strategy converges to a generalized uniform diversification strategy.
Highlights
Model uncertainty is a challenge that is inherent in many applications of mathematical models
We focus on ellipsoidal uncertainty sets K, see (4)
We show that the optimal strategy converges to a generalized uniform diversification strategy, where by uniform diversification we mean the equal weight or 1/d strategy for the investment in the risky assets
Summary
Model uncertainty is a challenge that is inherent in many applications of mathematical models. Pflug et al [21] study a one-period risk minimization problem under model uncertainty and show convergence of the optimal strategy to the uniform diversification strategy. Our results generalize these findings to a continuous-time utility maximization problem and provide an explanation for the good performance of the uniform diversification strategy in a continuous-time setting. Merton [16] solves this problem for power and logarithmic utility in a multivariate financial market model and gives a corresponding optimal strategy. To obtain strategies that are robust with respect to a possible misspecification of the drift we consider the worst-case optimization problem. 2 we state our multivariate, possibly incomplete, Black–Scholes type financial market model and introduce the robust utility maximization problem. If x ∈ Rd is a vector, x denotes the Euclidean norm of x
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