Abstract
We study a class of robust, or worst case scenario, optimal control problems for jump diffusions. The scenario is represented by a probability measure equivalent to the initial probability law. We show that if there exists a control that annihilates the noise coefficients in the state equation and a scenario which is an equivalent martingale measure for a specific process which is related to the control-derivative of the state process, then this control and this probability measure are optimal. We apply the result to the problem of consumption and portfolio optimization under model uncertainty in a financial market, where the price process S(t) of the risky asset is modeled as a geometric Ito-L00E9vy process. In this case the optimal scenario is an equivalent local martingale measure of S(t). We solve this problem explicitly in the case of logarithmic utility functions.
Mathematics Subject Classification (2010). 93E20, 60G51, 60H20, 60G44, 91G80.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
[AO] T.T.K. An and B. Oksendal: A maximum principle for stochastic differential games with partial information. J. Optim. Theory and Appl. 139 (2008), 463–483.
[BMS] G. Bordigoni, A. Matoussi and M. Schweizer: A stochastic control approach to a robust utility maximization problem. In F.E. Benth et al. (editors): Stochastic Analysis and Applications. Proceedings of the Second Abel Symposium, Oslo 2005. Springer 2007, pp. 135–151.
[HS] L.P. Hansen and T.J. Sargent: Robust control and model uncertainty. The American Economic Review, Vol. 91, no. 2, (2001), pp. 60–66.
[MO] S. Mataramvura and B. Oksendal: Risk minimizing portfolios and HJBI equations for stochastic differential games. Stochastics 80 (2008), 317–337.
[OS1] B. Oksendal and A. Sulem: Applied Stochastic Control of Jump Diffusions. Second Edition, Springer 2007.
[OS2] B. Oksendal and A. Sulem: A game theoretic approach to martingale measures in incomplete markets. Eprint, Dept. of Mathematics, University of Oslo 24/2006. Survey of Applied and Industrial Mathematics (TVP Publishers, Moscow), 15, (2008), 18–24.
[OS3] B. Oksendal and A. Sulem: Portfolio optimization under model uncertainty and BSDE games. Eprint, University of Oslo, Dept. of Mathematics, No. 1, 2011. Submitted to Quantitative Finance.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Basel AG
About this paper
Cite this paper
Øksendal, B., Sulem, A. (2011). Robust Stochastic Control and Equivalent Martingale Measures. In: Kohatsu-Higa, A., Privault, N., Sheu, SJ. (eds) Stochastic Analysis with Financial Applications. Progress in Probability, vol 65. Springer, Basel. https://doi.org/10.1007/978-3-0348-0097-6_12
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0097-6_12
Published:
Publisher Name: Springer, Basel
Print ISBN: 978-3-0348-0096-9
Online ISBN: 978-3-0348-0097-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)