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The 53rd ISCIE International Symposium on Stochastic Systems Theory and Its Applications

Special Lecture:
HJB Equations of Stochastic Control Problems with Exponential Type Criteria

Hideo Nagai (MMDS, Osaka University)


In risk-sensitive stochastic control, an exponential type criterion is formulated as the cost functional different from classical stochastic control. Then, to analyze its corresponding HJB equation including the risk-sensitive parameter, analysis according to the size of the parameter is required. Further, in establishing the verification theorem another difficulty arises coming from taking care of exponential (local) martingales. On the other hand, a small noise limit of risk-sensitive control is considered being H∞ control according to the idea by P. Whittle and mathematical analysis taking a singular limit of the HJB equation is necessary for its justification. Thus, we first introduce such specific features of studies of risk-sensitive control.

Similar problems to the above mentioned appear in discussing power utility maximization in mathematical finance for incomplete market models, which are often applied to practical issues as risk-sensitive investment management. Further, when discussing large deviation control which concerns the problem minimizing the down-side risk probability over large time, more delicate analysis of the HJB equation depending on the risk parameter may be necessary. If we consider the related problems to utility maximization admitting consumption, its criterion is formulated as the one of the exponential type combined with the classical one. We discuss mathematical analysis for such variants of the problems.


Hideo Nagai has been an Emeritus Professor, Osaka University since 2012. He was a Professor in the Division of Mathematical Science, Graduate School of Engineering Science at Osaka University from 1995 to 2012 and a Professor in the Department of Mathematics at Kansai University from 2012 to 2021. He is currently a Guest Professor in MMDS at Osaka University. He was a Research Associate in the Department of Mathematics at Tokyo Metropolitan University from 1976 to 1986 and a Visiting Researcher at INRIA Rocquencourt, France in 1984 for one year and in BiBoS at Bielefeld University, West Germany from Oct.1985 to Jan.1986. He received his Ph.D. Degree and M. Degree from Osaka University and completed undergraduate studies at Kyoto University. His research focuses on mathematical analysis related to stochastic control problems motivated by mathematical finance and engineering, based on the methods of stochastic analysis and PDE's ones.

He served as an Associate Editor for "Stochastics" (1995-2017), "International Journal of Stochastic Analysis" (2005-2017), "Applied Mathematics and Optimization" (2009-2016), and as an Editorial Board for "Asymptotic Analysis" (2000-2017). He has served as an Editorial Board for "Probability, Uncertainty and Quantitative Risk" (2015-) and an Honorary Editorial Board for "Stochastics" (2018-). He received Analysis Prize, The Mathematical Society of Japan in 2010.