$ G $-expectation approach to stochastic ordering

Abstract

<p style='text-indent:20px;'>This paper studies stochastic ordering under nonlinear expectations <inline-formula><tex-math id="M2">\begin{document}$ {\mathcal E}_{\mathcal{G}} $\end{document}</tex-math></inline-formula> generated by solutions of <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-Backward Stochastic Differential Equations (<inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula>-BSDEs) defined on <inline-formula><tex-math id="M5">\begin{document}$ G $\end{document}</tex-math></inline-formula>-expectation spaces. We derive sufficient conditions for the convex, increasing convex, and monotonic <inline-formula><tex-math id="M6">\begin{document}$ G $\end{document}</tex-math></inline-formula>-stochastic orderings of <inline-formula><tex-math id="M7">\begin{document}$ G $\end{document}</tex-math></inline-formula>-diffusion processes at terminal time. Our approach relies on comparison properties for <inline-formula><tex-math id="M8">\begin{document}$ G $\end{document}</tex-math></inline-formula>-Forward-Backward Stochastic Differential Equations (<inline-formula><tex-math id="M9">\begin{document}$ G $\end{document}</tex-math></inline-formula>-FBSDEs) and on relevant extensions of convexity, monotonicity and continuous dependence properties for the solutions of associated Hamilton-Jacobi-Bellman (HJB) equations. Applications of <inline-formula><tex-math id="M10">\begin{document}$ G $\end{document}</tex-math></inline-formula>-stochastic ordering to contingent claim superhedging price comparison under ambiguous coefficients are provided.</p>