Stable approximation of unbounded matrix operators for the simultaneous inversion of source terms and initial values in time-fractional Black–Scholes equation

This paper is devoted to identifying source term and initial value simultaneously in a time-fractional Black–Scholes equation, which is an ill-posed problem. The inverse problem is transformed into a system of unbounded operator equation system, and conditional stability is established under certain source conditions. We propose a stable approximation method to solve the problem, error estimates by rules of a priori and a posteriori regularization parameter selection are derived respectively. Numerical experiments are presented to validate the effectiveness of the proposed regularization method.