Regularization methods for identifying initial value of a time-fractional Black–Scholes equation

This paper investigates the initial value inversion of a time-fractional Black–Scholes equation, which is an ill-posed inverse problem. The quasi-reversibility regularization method and the Landweber iterative regularization method are employed to address this issue. Based on specific selection criteria for both a priori and a posteriori regularization parameters, we derive error estimates for the regularized solutions. The convergence rate obtained through the quasi-reversibility method is not universally optimal, whereas the convergence rate that is derived from the Landweber iterative approach consistently achieves order optimality. In our numerical experiments section, examples demonstrate that the proposed regularization methods are effective and stable.