Risk minimization in financial markets modeled by It?-Lévy processes

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• 作者：Bernt ?ksendal ; Agnès Sulem
• 关键词：Convex risk measure ; Risk minimization ; Recursive utility ; Utility optimization ; It? ; Lévy process ; Backward stochastic differential equation ; The maximum principle for stochastic control of FBSDE’s ; Stochastic differential game ; HJBI equation ; 60H10 ; 60H20 ; 60J75 ; 93E20 ; 91G80 ; 91G10 ; 91A23 ; 91B70 ; 91B30
• 刊名：Afrika Matematika
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：26
• 期：5-6
• 页码：939-979
• 全文大小：686 KB
• 参考文献：1.Aase, K.: Recursive utility and jump-diffusions. Manuscript, Norwegian School of Economics (NHH), 11 Dec 2013
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  15.?ksendal, B., Sulem, A.: Applied Stochastic Control of Jump Diffusions. 2nd edn. Springer (2007)
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• 作者单位：Bernt ?ksendal (1) (2)
  Agnès Sulem (1) (3) (4)
  
  1. Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316?, Oslo, Norway
  2. Norwegian School of Economics (NHH), Helleveien 30, 5045?, Bergen, Norway
  3. INRIA Paris-Rocquencourt, Domaine de Voluceau, BP 105, Le Chesnay Cedex, 78153, France
  4. Université Paris-Est, Marne-la-Vallée?, 77455, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics Education
  Applications of Mathematics
  History of Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：2190-7668

文摘

This paper is mainly a survey of recent research developments regarding methods for risk minimization in financial markets modeled by It?-Lévy processes, but it also contains some new results on the underlying stochastic maximum principle. The concept of a convex risk measure is introduced, and two representations of such measures are given, namely: (i) the dual representation and (ii) the representation by means of backward stochastic differential equations (BSDEs) with jumps. Depending on the representation, the corresponding risk minimal portfolio problem is studied, either in the context of stochastic differential games or optimal control of forward-backward SDEs. The related concept of recursive utility is also introduced, and corresponding recursive utility maximization problems are studied. In either case the maximum principle for optimal stochastic control plays a crucial role, and in the paper we prove a version of this principle which is stronger than what was previously known. The theory is illustrated by examples, showing explicitly the risk minimizing portfolio in some cases. Keywords Convex risk measure Risk minimization Recursive utility Utility optimization It?-Lévy process Backward stochastic differential equation The maximum principle for stochastic control of FBSDE’s Stochastic differential game HJBI equation