{
  "id": "1405.0251",
  "version": "v3",
  "published": "2014-05-01T19:01:52.000Z",
  "updated": "2014-09-09T13:37:36.000Z",
  "title": "Robust utility maximization without model compactness",
  "authors": [
    "Julio Backhoff",
    "Joaquín Fontbona"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "We formulate conditions for the solvability of the problem of robust utility maximization from final wealth in continuous time financial markets, without assuming weak compactness of the densities of the uncertainty set. Relevant examples of such a situation typically arise when the uncertainty set is determined through moment constraints, the treatment of which is beyond the reach of the existing literature. Our approach is based on identifying functional spaces naturally associated with the elements of each problem. For general markets these are modular spaces, through which we can prove a minimax equality and the existence of optimal strategies by exploiting the compactness, which we establish, of the image by the utility function of the set of attainable wealths. This line of argumentation starkly differs from the usual ones in the literature, in particular by not requiring a worst-case measure to exist. In complete markets the relevant space is an Orlicz space, and upon granting its reflexivity under verifiable conditions on the utility function, we obtain additionally the existence of a worst-case measure. Combining our ideas with abstract entropy minimization techniques, we moreover provide in that case a novel methodology to characterize such measure. This is done in terms of the solution to a certain \"bi-dual\" problem, which can in practical cases be simpler to solve.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-09T09:23:23.000Z",
      "abstract": "In the robust utility maximization problem, and agent wishes to maximize her expected utility from terminal wealth under the worst possible probabilistic model in a fixed uncertainty set, which we suppose dominated by a reference measure. In this work we aim at relaxing the usual compactness assumptions on the set of densities thereof, by identifying relevant Banach spaces where a fortiori worst-case measures (when they exist) should live and formulating conditions on these spaces for the solvability of the original problem. In complete markets the relevant space is an Orlicz space and upon granting its reflexivity (which we can do under simple assumptions on the utility function) we obtain attainability of a \\textit{worst-case measure} and optimal strategies. Furthermore, by means of entropy minimization techniques we can give an explicit characterization of this measure in terms of the solution to a certain dual problem which in principle can be easier to solve. For general markets, we show that the relevant Banach space is a certain Modular space which, no matter the ingredients of the problem, is practically never (beyond the complete case) a reflexive space. Nevertheless, we can still prove in the general setting a minimax equality and the existence of optimal strategies, without resorting to model compactness assumptions nor ensuring existence of a worst-case measure.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-09-09T13:37:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "91G10",
      "49N15",
      "46E30"
    ],
    "keywords": [
      "optimal strategies",
      "robust utility maximization problem",
      "worst-case measure",
      "model compactness assumptions",
      "usual compactness assumptions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.0251B"
    }
  }
}