2007 Fiscal Year Final Research Report Summary
Expected utility maximiaation problems and stochastic control
| Project/Area Number |
16340025
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Osaka University |
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Principal Investigator |
NAGAI HIideo Osaka University, graduate school of engineering science, professor (70110848)
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| Co-Investigator(Kenkyū-buntansha) |
KOHATSU-HIGA Artuto Osaka University, graduate school of engineering science, associate professor (80420412)
KOTANI Shinichi Osaka university, graduate school ofscience, professor (10025463)
MATSUMOTO Hiroyuki Nagoya university, graduate schcolofinformationscience, professor (00190538)
ISHII Hitoshi Waseda University, Schealofeducationand integrated sciences, Professor (70102887)
KOIKE Shigeaki Saitama university, graduate schcolofscience and technology, professor (90205295)
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| Project Period (FY) |
2004 – 2007
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| Keywords | expected utility maximization problems / quasi-variational inequalities / transaction costs / insider trading / viscosity solutions / Hamilton-Jacobi equation / large deviation control |
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| Research Abstract |
We considered expected power utility maximization problems on infinite time horizon with transaction costs. Related risk-sensitive quasi-variational inequalities were deduced. The solution of the inequality consists of the pair of constant and a function. We constructed an optimal strategy by using the function of the solution and showed the constant give the optimal value of the problem.
We studied problems on equilibrium price for insider trading and showed that there exists a stable equilibrium price for such models even though insiders affect stock prices.
We studied asymptotic behavior of the solution to a Hamilton-Jacobi equation and showed that the solution converges to the asymptotic solution on the whole. Euclidean space under relatively general assumptions.
By introducing the notion of Lp viscosity solutions we proved the existence of Lp viscosity solutions by a modified version of the Perron's method. Moreover we showed his solution turns out to be Holder continuous if the equation is uniform elliptic.
By proving differentiabilities of viscosity solutions of the obstacle problems arising from mathematical finance we constructed optimal controls.
By taking up linear Gaussian models which are incomplete market models we considered the problem minimizing a probability that growth rate of the wealth process lies below the prescribed value. The asymptotics of the probability is characterized as the dual of risk-sensitive portfolio optimization problem.
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