| Abstract: |
| We introduce the following linear stochastic equation of Schr{\o}dinger type
\[dX(t,x) = (1+a\,i)\Delta_xX(t,x)dt + \sigma\, i\sum^{\infty}_{k=1}\mu_k (X(t,\cdot),h_k(\cdot))_Hh_k(x)dB^h_k(t),\]
with $x\in[0,1]$, $X(0)=X_0 \in H=L^2[0,1]$, $X(t,0)=X(t,1)=0$. $(B^h_k(t))_{t\geq 0}$, ($k=1,2,\ldots$) denote independent fractional Brownian motions with Hurst index $h\in ]1/2,1[$. Let $h_k(\cdot)$ $(k=1,2,\ldots)$ be the eigenfunctions of Laplacian operator with above boundaries. Finally, let be $\mu_k \in R^1$ with $\sum^{\infty}_{k=1}\mu_k^2 < \infty$. The solution process is defined in mild sense and is approximated by the Galerkin approximation. We get a sequence of one dimensional problems.\
A mean square estimation criterion is used for
these equations to estimate $a$. It is proved that the estimation is unbiased and weakly consistent for the original problem. |
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