Optimal interpolation of random Gaussian field with non-zero mean (Q2703328)

Observe the random field \(\xi_{z}, z=(z_{1},z_{2})\in [0,T]= [0,T_{1}]\times[0,T_{2}]\) and \(\xi_{z}=\int_{[0,z]} (A(u)\theta_{u}+B(u))du+\int_{[0,z]}C(u)dw_{u}\), where \(\theta_{z}\) is a non-observed field with non-zero mean; \(w_{z}\) is a Wiener field; \(A(u)\), \(B(u)\), \(C(u)\) are known non-random functions. Let \(\theta_{z}\) satisfy the stochastic Goursat problem NEWLINE\[NEWLINE{\partial^{2}\theta_{z}\over \partial z_{1}\partial z_{2}}-\alpha(z){\partial\theta_{z}\over\partial z_{1}}-\beta(z){\partial\theta_{z}\over\partial z_{2}}+ \gamma(z)\theta_{z}=a(z)+b(z)\ddot w_{z}',NEWLINE\]NEWLINE \(\theta_{z}=0\) if \(z_{1}=0\) or \(z_{2}=0\), where \(\alpha(z),\beta(z),\gamma(z), a(z), b(z)\) are known non-random functions; \(w_{z}'\) is a Wiener field independent on \(w_{z}\). For the optimal estimate \(\overline\theta_{z}=E(\theta_{z}\mid {\mathcal F}_{T}^{\eta})\), \({\mathcal F}_{T}^{\eta}=\sigma\{\eta_{z}, z\in[0,T]\}\) the authors obtain a stochastic partial differential equation with boundary value conditions.