Remarks on numerical approximation of Volterra integral equations by Walsh–Hadamard transform

Walsh functions form a piecewise-constant orthonormal basis that is particularly well-suited for digital computation and signal approximation. Nevertheless, the direct evaluation of Walsh transforms for discrete functions becomes computationally prohibitive as the resolution increases. To overcome this difficulty, we develop an efficient numerical scheme based on the Fast Walsh–Hadamard–Fourier Transform (FWHFT) for the approximation of solutions to Volterra integral equations. The proposed method reduces the computational complexity from O(2²ⁿ) to O(n2ⁿ), thereby rendering the approach scalable to high-resolution problems. We present a complete algorithmic framework that exploits this fast transform and analyze its performance on a variety of examples. In particular, we illustrate several examples for the broad applicability of the method. These examples highlight the principal advantages of the FWHFT approach, as well as certain limitations inherent in the transform structure. As a further application, we implement the method in a financial setting by addressing the problem of pricing European options under the Bachelier model. This example demonstrates not only the accuracy of the proposed algorithm but also its practical relevance to computational finance, especially in scenarios involving structured payoff functions. Numerical experiments confirm the expected convergence behavior and the substantial computational savings afforded by the method. Finally, we discuss possible extensions of the approach to fractional-order models, which are naturally linked to Volterra-type integral equations and arise frequently in applications.