Abstract
We state sufficient conditions for the uniqueness of minimizers of the Tikhonov functional corresponding to the local volatility surface calibration problem. We use two different approaches. In the first one, using smoothness properties of the forward operator, we prove that, if the regularization parameter is sufficiently large, then, uniqueness hold. Secondly, we find sufficient conditions for the local uniqueness of Tikhonov minimizers by analyzing the geometrical properties of the image through the forward operator of line segments. We also study some theoretical consequences, such as, convergence-rate results with respect to the noise level, as well as, the well-posedness of an a posteriori rule for choosing the regularization parameter. In addition, we provide some numerical examples.
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