Abstract
This paper is mainly a survey of recent research developments regarding methods for risk minimization in financial markets modeled by Itô-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.
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Acknowledgments
These lecture notes are based on our earlier works and lectures given on this topic, and on a course on risk minimization that B.Ø. gave at NHH, in 2013. We are grateful to K. Aase, J. Haug, S.-A. Persson and J. Ubøe for valuable comments.
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The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no [228087].
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Øksendal, B., Sulem, A. Risk minimization in financial markets modeled by Itô-Lévy processes. Afr. Mat. 26, 939–979 (2015). https://doi.org/10.1007/s13370-014-0248-9
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DOI: https://doi.org/10.1007/s13370-014-0248-9
Keywords
- Convex risk measure
- Risk minimization
- Recursive utility
- Utility optimization
- Itô-Lévy process
- Backward stochastic differential equation
- The maximum principle for stochastic control of FBSDE’s
- Stochastic differential game
- HJBI equation