Robust Utility Maximization in a Multivariate Financial Market With Stochastic Drift

Abstract

We study a utility maximization problem in a financial market with a stochastic drift process, combining a worst-case approach with filtering techniques. Drift processes are difficult to estimate from asset prices, and at the same time optimal strategies in portfolio optimization problems depend crucially on the drift. We approach this problem by setting up a worst-case optimization problem with a time-dependent uncertainty set for the drift. Investors assume that the worst possible drift process with values in the uncertainty set will occur. This leads to local optimization problems, and the resulting optimal strategy needs to be updated continuously in time. We prove a minimax theorem for the local optimization problems and derive the optimal strategy. Further, we show how an ellipsoidal uncertainty set can be defined based on filtering techniques and demonstrate that investors need to choose a robust strategy to be able to profit from additional information.

Robust utility maximizing strategies under model uncertainty and their convergence

“…Björk et al [3], in our original formulation (3). But for the special case of (11) with a constant uncertainty set K , the results in Sass and Westphal [24] show that one obtains at time t the same optimal risky fractions as when starting at time 0. In combination with the Bellman principle, which implies that at time t we only need the information X t = x, this proves that our robust utility maximization problem with optimal solution π * obtained in Sect.…”

“…Naturally, the optimal strategy of the investor will then also be adapted continuously. In Sass and Westphal [24] it is shown in detail how the results of this paper can be used to solve the above described more complicated problem. An explicit representation of the optimal strategy and a minimax theorem can be derived.…”

“…This flexibility in the form of Γ is especially useful for a generalization of our model to a setting with timedependent drift and uncertainty sets, see Sect. 5, where we give a short outlook on Sass and Westphal [24]. In that follow-up work a time-dependent uncertainty set is constructed based on filtering techniques.…”

“…In this section we want to give a brief outlook on how the results of this paper can be applied also in more general financial market models with a stochastic drift process. This generalization is the topic of our follow-up work Sass and Westphal [24]. Here we only give a short outline of the setup to illustrate the relevance of this work.…”

“…In Sass and Westphal [24] the results of the present paper are generalized to a financial market with a stochastic drift process and time-dependent uncertainty sets (K t ) t∈[0,T ] . This is motivated by the idea that information about the hidden drift process, as e.g.…”