2001 Fiscal Year Final Research Report Summary
Asymptotic statistical Inference theory for stochastic process
| Project/Area Number |
11680319
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Statistical science
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| Research Institution | The University of Tokyo |
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Principal Investigator |
YOSHIDA Nakahiro Graduate School of Mathematic Sciences, The University of Tokyo, Associate Professor, 大学院・数理科学研究科, 助教授 (90210707)
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| Project Period (FY) |
1999 – 2001
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| Keywords | asymptotic expansion / Malliavin calculus / filtering / partial mixing / derivative / information criterion / conditional expectation / long-memory process |
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| Research Abstract |
1. We presented an asymptotic expansion to a continuous-time Markov process satisfying the mixing condition. The conditional type Cramer condition in discrete-time setting was replaced by the nondegeneracy condition of the Malliavin covariance of the functional. This method applied to stochastic differential equations. The validity problem was at the same time solved.
2. Result 1 was applied to statistical parametric models, and expansions for M-estimators were derived. For diffusion models, this paper completely described the coefficients in the formulae. This, together with Result 1 and Result 5 below, has been the fundamental literature in this field, and it is applied to various statistical problems today.
3. For statistical models of diffusion processes with small noises, we derived so-called information criteria for model selection.
4. When the security price is described by a general nonlinear stochastic differential equation, it is a difficult problem to compute the value of a der
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ivative. However, it is possible to approximate it by means of the asymptotic expansion technique. This method was introduced by the author and recently many authors pursuit this method. Result 4 treated a practical situation, that is, the equation of the security has unknown parameters. This paper assessed the effects of substitution of estimators to the approximation of option prices, and proposed a correct way to the good approximation.
5. Since for a stochastic differential equation with random coefficients of long memory, the usual mixing condition was brokenn, a new methodology is necessary to derive asymptotic expansions. Result 5 introduced the notion of partial mixing and successfully derived asymptotic expansions (of course with validity). Moreover, it showed a practical convenient method with the support theorem to verify the local nondegeneracy of the Malliavin covariance of the expanded functional. This new device enables us to derive expansions very easily, like i.i.d. models.
6. Conditional expectation is a most irregular functional in limit theorems. We applied the Malliavin calculus to derive asymptotic expansion of the conditional expectation. This result was applied to the stochastic differential equation with jumps and filtering problems. Less
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Research Products
(12 results)
