Dutch Book Methods for Difference and Differential Equations

Alberto Gandolfi, Jianhan Hu

Research output: Chapter in Book/Report/Conference proceedingChapter


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.

Original languageEnglish (US)
Title of host publicationTrends in Mathematics
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages11
StatePublished - 2024

Publication series

NameTrends in Mathematics
VolumePart F2357
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X


  • 12D99
  • 34B99
  • 39A05
  • 60A99
  • Boundary value problems
  • Difference equations
  • Differential equations
  • Dutch Book
  • Existence of solutions
  • Fredholm alternative
  • Positivstellensatz

ASJC Scopus subject areas

  • General Mathematics


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