Special Lecture: HJB Equations of Stochastic Control Problems with Exponential Type Criteria
Hideo Nagai (MMDS, Osaka University)
Abstract
In risksensitive 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 risksensitive
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 risksensitive 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 risksensitive 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 risksensitive investment management.
Further, when discussing large deviation control which concerns the problem
minimizing the downside 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.
Biography
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" (19952017),
"International Journal of Stochastic Analysis" (20052017),
"Applied Mathematics and Optimization" (20092016), and as an Editorial Board
for "Asymptotic Analysis" (20002017).
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.
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