$ G $-expectation approach to stochastic ordering

Sel Ly; Nicolas Privault

doi:10.3934/fmf.2021012

Article Contents

2022, Volume 1, Issue 3: 343-374. Doi: 10.3934/fmf.2021012 open access

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$ G $-expectation approach to stochastic ordering

Sel Ly and
Nicolas Privault^,

School of Physical and Mathematical Sciences, Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore

^* Corresponding author: Nicolas Privault
^* Corresponding author: Nicolas Privault

Received: December 2020

Revised: August 2021

Early access: May 2022

Published: September 2022

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Abstract

This paper studies stochastic ordering under nonlinear expectations $ {\mathcal E}_{\mathcal{G}} $ generated by solutions of $ G $-Backward Stochastic Differential Equations ($ G $-BSDEs) defined on $ G $-expectation spaces. We derive sufficient conditions for the convex, increasing convex, and monotonic $ G $-stochastic orderings of $ G $-diffusion processes at terminal time. Our approach relies on comparison properties for $ G $-Forward-Backward Stochastic Differential Equations ($ G $-FBSDEs) and on relevant extensions of convexity, monotonicity and continuous dependence properties for the solutions of associated Hamilton-Jacobi-Bellman (HJB) equations. Applications of $ G $-stochastic ordering to contingent claim superhedging price comparison under ambiguous coefficients are provided.

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Mathematics Subject Classification: Primary: 60E15; Secondary: 35B51, 60H10, 60H30.

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