Geodesic
Estimation problems for coefficients of stochastic partial differential equations. Part~I
Teoriâ veroâtnostej i ee primeneniâ, Tome 43 (1998) no. 3, pp. 417-438

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This paper considers the problem of estimating functional parameters $a_k(t,x)$, $f(t,x)$ by observing a solution $u_{\varepsilon}(t,x)$ of a stochastic partial differential equation $$ du_{\varepsilon}(t)=\sum_{|k|\le2p}a_kD_x^ku_{\varepsilon}+f\,dt+\varepsilon\,dw(t), $$ where $w(t)$ is a Wiener process. The asymptotic statement of the problem is considered when the noise level $\varepsilon\to0$. In the first part of the work we determine what is considered the statistics of the problem and investigate the problem of estimating $f$.
Keywords: keywords inverse problems, stochastic partial differential equations, statistical estimating, nonparametric estimating problems. 
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     author = {I. A. Ibragimov and R. Z. Khas'minskii},
     title = {Estimation problems for coefficients of stochastic partial differential equations. {Part~I}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {417--438},
     publisher = {mathdoc},
     volume = {43},
     number = {3},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1998_43_3_a0/}
}
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I. A. Ibragimov; R. Z. Khas'minskii. Estimation problems for coefficients of stochastic partial differential equations. Part~I. Teoriâ veroâtnostej i ee primeneniâ, Tome 43 (1998) no. 3, pp. 417-438. http://geodesic.mathdoc.fr/item/TVP_1998_43_3_a0/