Appl Math Optim 37:43–69 (1998)
1998 Springer-Verlag New York Inc.
Identiﬁcation of a Discontinuous Parameter in
Stochastic Parabolic Systems
S. I. Aihara
Department of Management and Systems Science,
The Science University of Tokyo, Suwa College,
5000-2 Toyohira, Chino 391-02, Nagano, Japan
Communicated by A. Bensoussan
Abstract. The purpose of this paper is to study the identiﬁcation problem for
a spatially varying discontinuous parameter in stochastic diffusion equations. The
consistency property of the maximum likelihood estimate (M.L.E.) and a generating
algorithm for M.L.E. have been explored under the condition that the unknown
parameter is in a sufﬁciently regular space with respect to spatial variables. In
order to prove the consistency property of the M.L.E. for a discontinuous diffusion
coefﬁcient, we use the method of sieves, i.e., ﬁrst the admissible class of unknown
parameters is projected into a ﬁnite-dimensional space and next the convergence of
the derived ﬁnite-dimensional M.L.E. to the inﬁnite-dimensional M.L.E. is justiﬁed
under some conditions. An iterative algorithm for generating the M.L.E. is also
proposed with two numerical examples.
Key Words. Maximum likelihood estimate, Consistent estimate, The method of
sieves, Stochastic parabolic equations.
AMS Classiﬁcation. Primary 93C20, 93E12, Secondary 60H15.
This paper deals with a problem of identifying discontinuous diffusion coefﬁcients in
stochastic diffusion equations under noisy partial observations. In stochastic systems, the
This research was partially supported by the Japan Ministry of Education under Grant C-05650423.