Dutch Book Methods for Difference and Differential Equations

Based on gambling setups in which agents have positive gains at no risk, Dutch Book Arguments appear in various contexts such as foundations of probability theory, economics, and philosophy as certificates of inconsistency. In this paper, we introduce the concept of Dutch Book Argument for some classes of difference and ordinary differential equations with various types of mixed boundary conditions with the intent of determining conditions for the existence of solutions. To set up a Dutch Book witnessing absence of solutions, we exploit various tools: for linear difference equations, we use the adjoint equation; for difference equations with polynomial terms, we exploit the Positivstellensatz; and for linear differential equations, we first use Fredholm alternative principle, and then define and employ an extension of it. Although this is a very preliminary step, we hope that the Dutch Book Arguments in differential equations can be extended to nonlinear, e.g., polynomial, cases.