[Paper Review] Integral equations for Rost's reversed barriers: existence and uniqueness results

Tiziano De Angelis, Yerkin Kitapbayev|arXiv (Cornell University)|Aug 24, 2015

Stochastic processes and financial applications43 references4 citations

TL;DR

This paper establishes that Rost's reversed barriers in the Skorokhod embedding problem are uniquely characterized as the solution to a system of nonlinear Volterra integral equations of the second kind. Using purely probabilistic methods, the authors prove existence and uniqueness of the left-continuous monotonic boundaries $s_+$ and $s_-$ for atomless target distributions $\mu$, extending beyond prior PDE and constructive approaches by providing a direct, numerically tractable characterization via integral equations involving general probability measures.

ABSTRACT

We establish that the boundaries of the so-called Rost's reversed barrier are the unique couple of left-continuous monotonic functions solving a suitable system of nonlinear integral equations of Volterra type. Our result holds for atom-less target distributions $\\mu$ of the related Skorokhod embedding problem. The integral equations we obtain here generalise the ones often arising in optimal stopping literature and our proof of the uniqueness of the solution goes beyond the existing results in the field.

Motivation & Objective

• To provide a direct probabilistic characterization of Rost’s reversed barriers in the Skorokhod embedding problem.
• To establish the existence and uniqueness of the boundaries $s_+$ and $s_-$ as solutions to a system of nonlinear Volterra integral equations.
• To offer a computationally efficient alternative to existing PDE-based or constructive approximation methods for computing these barriers.
• To extend the scope of integral equation theory by handling general probability measures $\mu$ and $\nu$ rather than just Lebesgue-dense measures.
• To resolve open questions on solvability and uniqueness of integral equations in optimal stopping and free-boundary problems.

Proposed method

• Derive a system of nonlinear Volterra integral equations of the second kind that characterize the boundaries $s_+$ and $s_-$ of Rost’s reversed barriers.
• Use the connection between Rost’s solution and an optimal stopping problem to frame the boundary characterization as a solution to integral equations involving local time and hitting probabilities.
• Apply stochastic calculus and probabilistic coupling arguments to prove existence and uniqueness, avoiding PDE or viscosity solution techniques.
• Transform the time-horizon problem via time reversal to define $b_\pm(t) = s_\pm(T - t)$, enabling recursive numerical solution.
• Develop a numerical algorithm based on piecewise constant approximations of the boundaries and iterative solution of algebraic equations at each time step.
• Use a step-by-step recursive scheme with time discretization $t_k = kh$, solving equations (4.1) and (4.2) at each $k$ using Newton-Raphson or bisection methods.

Experimental results

Research questions

• RQ1 Can the boundaries of Rost’s reversed barriers be uniquely characterized as solutions to a system of integral equations without relying on PDE or approximation schemes?
• RQ2 Do the integral equations derived here generalize existing Volterra equations in optimal stopping theory to include general probability measures $\mu$ and $\nu$?
• RQ3 Is the solution to these integral equations unique under the assumption of atomless $\mu$, even when $\mu$ is singular (e.g., Cantor distribution)?
• RQ4 Can a numerically stable and convergent algorithm be constructed directly from the integral equations without solving variational inequalities?
• RQ5 What is the theoretical justification for the convergence of the proposed numerical scheme as the time step $h \to 0$?

Key findings

• The boundaries $s_+$ and $s_-$ are the unique left-continuous monotonic solutions to the system of nonlinear Volterra integral equations (2.11), valid for any atomless target distribution $\mu$.
• The proof of uniqueness relies on a novel probabilistic argument involving local time and coupling, going beyond existing results in optimal stopping and integral equation theory.
• The integral equations generalize classical forms by involving a product measure $dt \times (\nu - \mu)(dx)$, allowing for singular or discrete components in $\mu$ and $\nu$, not just densities.
• The proposed numerical algorithm converges to the true boundaries as the time step $h \to 0$, with convergence empirically confirmed in Figures 2 and 3.
• The method avoids the need for solving variational inequalities or constructing piecewise constant approximations, offering a direct and computationally efficient alternative to Cox-Peskir and PDE-based approaches.
• The results are robust even for singular distributions like the Cantor distribution, as long as $\mu$ is atomless, demonstrating the generality of the framework.