Numerical Methods for Integral Equations of the Second Kind With Non-Smooth Solutions of Bounded Variation

This paper develops a finite approximation approach to find the solution $F(x)$ of integral equations in the form $F(x) = \xi(x)+\int_\mathbb{R} \kappa(x,u) dF(u),\forall x\in\mathbb{R}$, where the unknown distributional solution $F(x)$ defines the measure associated with the integration, and $F(x)$ may not be continuous. To the best of our knowledge, the equation solutions in this framework have never been studied before. However, such equations arise frequently when modeling stochastic systems. We construct a Banach space of (right-continuous) distribution functions and reformulate the problem into an operator equation. We provide general necessary and sufficient conditions that allow us to show convergence of the approximation approach developed in this paper. We then provide two specific choices of approximation sequences and show that the properties of these sequences are sufficient to generate approximate equation solutions that converge to the true solution assuming solution uniqueness and some additional mild regularity conditions. Our analysis is performed under the supremum norm, allowing wider applicability of our results. Worst-case error bounds are also available from solving a linear program. We demonstrate the viability and computational performance of our approach by constructing three examples. The solution of the first example can be constructed manually, but demonstrates the correctness and convergence of our approach. The second application example involves stationary distribution equations of a stochastic model, and demonstrates the dramatic improvement our approach provides over the use of computer simulation. The third example solves a problem involving an everywhere non-differentiable function for which no closed-form solution is available