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Asymptotic statistical inference for a stochastic heat flow problem. (English) Zbl 0589.62075

This note demonstrates how the framework of statistical inference for Hilbert space valued stochastic differentials (SDE’s) established by the second author [Stochastic Processes Appl. 17, 243-263 (1984; Zbl 0553.93059)] is made to apply in certain cases, where the parameter of interest is contained in an unbounded operator generating a semigroup on the Hilbert space. In particular, the results are relevant in the context of asymptotically effective statistical inference about surface conductance parameters in a stochastic heat flow given a direct, noise- free measurement of the sample path.

MSC:

62M05 Markov processes: estimation; hidden Markov models
60B11 Probability theory on linear topological spaces
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
47D03 Groups and semigroups of linear operators
62F12 Asymptotic properties of parametric estimators
62P99 Applications of statistics

Citations:

Zbl 0553.93059
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References:

[1] Akritas, M. G.; Johnson, R. A., The contiguity of probability measures and asymptotic inference in continuous time stationary diffusion and Gaussian processes with known covariance, Journal of Multivariate Analysis, 22, 123-135 (1984) · Zbl 0526.62079
[2] Basawa, I. V.; Rao, L. S., Statistical Inference for Stochastic Processes (1980), Academic Press: Academic Press London · Zbl 0448.62070
[3] Carslaw, H. S.; Jaeger, J. C., Conduction of Heat in Solids (1959), Oxford University Press: Oxford University Press Oxford · Zbl 0972.80500
[4] Curtain, R. F.; Pritchard, A. J., Infinite Dimensional Linear Systems Theory (1978), Springer-Verlag: Springer-Verlag Berlin · Zbl 0426.93001
[5] Liptser, R. S.; Shiryayev, A. N., Statistics of Random Processes I (1977), Springer Verlag: Springer Verlag Berlin · Zbl 0364.60004
[6] Loges, W., Girsanov’s theorem in Hilbert space and an application to the statistics of Hilbert space valued stochastic differential equations, Stochastic Processes and their Applications, 17, 243-263 (1984) · Zbl 0553.93059
[7] Zabczyk, J., Structural properties and limit behaviour of linear stochastic systems in Hilbert spaces, (Mathematical Control Theory, Banach Center Publications, 13 (1983), Polish Scientific Publishers: Polish Scientific Publishers Warsaw) · Zbl 0573.93076
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