Science in China Series A: Mathematics

, Volume 52, Issue 6, pp 1157–1168

On nonlinear ill-posed inverse problems with applications to pricing of defaultable bonds and option pricing


    • Department of EconomicsYale University
    • The Guanghua School of ManagementPeking University
    • School of EconomicsShanghai University of Finance and Economics
  • Demian Pouzo
    • Department of EconomicsNew York University

DOI: 10.1007/s11425-009-0058-y

Cite this article as:
Chen, X. & Pouzo, D. Sci. China Ser. A-Math. (2009) 52: 1157. doi:10.1007/s11425-009-0058-y


This paper considers the estimation of an unknown function h that can be characterized as a solution to a nonlinear operator equation mapping between two infinite dimensional Hilbert spaces. The nonlinear operator is unknown but can be consistently estimated, and its inverse is discontinuous, rendering the problem ill-posed. We establish the consistency for the class of estimators that are regularized using general lower semicompact penalty functions. We derive the optimal convergence rates of the estimators under the Hilbert scale norms. We apply our results to two important problems in economics and finance: (1) estimating the parameters of the pricing kernel of defaultable bonds; (2) recovering the volatility surface implied by option prices allowing for measurement error in the option prices and numerical error in the computation of the operator.


nonlinear ill-posed inverse problemsHilbert Scalesoptimal convergence ratespricing of defaultable bondsoption prices



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© Science in China Press and Springer-Verlag GmbH 2009