New Applied Mathematics Findings Reported from Sgh Warsaw School Economics (Asymptotic Optimality of a First-order Approximate Strategy for an Exponential Utility Maximization Problem With a Small Coefficient of Wealth-dependent Risk Aversion)

2021 AUG 12 (NewsRx) -- By a News Reporter-Staff News Editor at Insurance Daily News -- A new study on Mathematics - Applied Mathematics is now available. According to news reporting originating from Warsaw, Poland, by NewsRx editors, the research stated, “In Delong [8] we investigate an exponential utility maximization problem for an insurer who faces a stream of non-hedgeable claims. We assume that the insurer’s risk aversion coefficient consists of a constant risk aversion and a small amount of wealth-dependent risk aversion.”

Financial support for this research came from Szkola Glowna Handlowa w Warszawie.

Our news editors obtained a quote from the research from Sgh Warsaw School Economics, “We apply perturbation theory and expand the equilibrium value function of the optimization problem on the parameter epsilondocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$epsilon $$end{document} controlling the degree of the insurer’s risk aversion depending on wealth. We derive a candidate for the first-order approximation to the equilibrium investment strategy.”

According to the news editors, the research concluded: “In this paper we formally show that the zeroth-order investment strategy pi 0*documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$pi _0 <. >*$$end{document} postulated by Delong (Math Methods Oper Res 89:73-113, 2019) performs better than any strategy pi 0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$pi _0$$end{document} when we compare the asymptotic expansions of the objective functions up to order O(1)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {O}}(1)$$end{document} as epsilon -> 0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$epsilon rightarrow 0$$end{document}, and the first-order investment strategy pi 0*+pi 1*epsilondocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$pi _0 <. >*+pi _1 <. >*epsilon $$end{document} postulated by Delong (Math Methods Oper Res 89:73-113, 2019) is the equilibrium strategy in the class of strategies pi 0*+pi 1 epsilondocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$pi <. >*_0+pi _1epsilon $$end{document} when we compare the asymptotic expansions of the objective functions up to order O(epsilon 2)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${mathcal {O}}(epsilon <. >2)$$end{document} as epsilon -> 0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$epsilon rightarrow 0$$end{document}, where epsilondocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$epsilon $$end{document} denotes the parameter controlling the degree of the insurer’s risk aversion depending on wealth.”

This research has been peer-reviewed.

For more information on this research see: Asymptotic Optimality of a First-order Approximate Strategy for an Exponential Utility Maximization Problem With a Small Coefficient of Wealth-dependent Risk Aversion. Applied Mathematics & Optimization, 2021;84(1):649-682. Applied Mathematics & Optimization can be contacted at: Springer, One New York Plaza, Suite 4600, New York, Ny, United States. (Springer - www.springer.com; Applied Mathematics & Optimization - http://www.springerlink.com/content/0095-4616/)

The news editors report that additional information may be obtained by contacting Lukasz Delong, Sgh Warsaw School Economics, Coll Econ Anal, Econometrics Institute, Niepodleglosci 162, Pl-02554 Warsaw, Poland.

The direct object identifier (DOI) for that additional information is: https://doi.org/10.1007/s00245-020-09657-4. This DOI is a link to an online electronic document that is either free or for purchase, and can be your direct source for a journal article and its citation.

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