Uncertainty orders on the sublinear expectation space

Tian, Dejian; Jiang, Long

doi:10.1515/math-2016-0023

Cited by 3 publications

(4 citation statements)

References 21 publications

(33 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A stochastic ordering via sublinear expectations has also been defined in [92] by combining (4.9) with the inequality…”

Section: Comparison Results For G-risk Measuresmentioning

confidence: 99%

“…Stochastic orderings with respect to capacity have been studied in [37,36] by using Choquet's expectation and uncertainty orders have been constructed in [92] on the sublinear expectation space. Here, we extend this approach to the comparison of random variables X (1) , X (2) in the settings of g-expectations and g-evaluations, which are not sublinear in general, via the condition…”

Section: G-stochastic Orderingsmentioning

confidence: 99%

“…In this chapter, we go beyond the linear setting of the standard expectation operator IE, and we study stochastic orderings from the point of view of nonlinear expectations, which leads to more general concepts of stochastic ordering in nonadditive models. Uncertainty orders have been constructed in [92] on the sublinear expectation space. Here, we extend this approach to the settings of g-expectations and g-evaluations, which are not sublinear in general.…”

Section: Introductionmentioning

confidence: 99%

“…The goal of this chapter is to study stochastic orderings under a nonlinear Gexpectation framework, which leads to more general concepts of stochastic orderings in comparison with the standard linear expectation setting. Uncertainty orders of G-normal, G-maximal and G-distributed random variables have been studied in [92] on the sublinear expectation space. In addition, in Chapter 4, a stochastic ordering has also been studied under the g-expectations and g-evaluations.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Stochastic orderings by nonlinear expectations

Ly¹

View full text Add to dashboard Cite

We study the theory of stochastic order under the nonlinear expectations framework, including g-and G-expectations, which leads to more general concepts of orderings in comparison with the standard linear expectation setting. The g-expectation E g is generated by a solution of Backward Stochastic Differential Equation (BSDE) and it provides a new robust tool for option pricing in finance. In the same manner, we denote by E G a nonlinear expectation generated by a solution of G-Backward Stochastic Differential Equation (G-BSDE), which is defined on G-expectation spaces. In this dissertation, our main contributions consist of four parts as follows:(i) We extend several properties of monotonicity, convexity and continuous dependence for the solutions of Forward-Backward Stochastic Differential Equations (FBSDEs) and associated semilinear parabolic Partial Differential Equations (PDEs). Also, we obtain analogous results for the solutions of G-Forward-Backward Stochastic Differential Equations (G-FBSDEs) and associated G-PDEs (Hamilton-Jacobi-Bellman-type equations). These properties are very crucial to constitute comparison results for the monotonic, convex, increasing convex g-and G-stochastic orderings.(ii) We give some characterizations for a better understanding of g-stochastic orderings which are defined via the g-evaluations and g-expectations E g . We then derive several sufficient conditions for the convex, increasing convex and monotonic g-stochastic orderings of diffusion processes at the terminal time T . Analogous comparison results for g-risk measures have been proposed as consequences, in terms of concave g-stochastic orderings. Applications of the g-stochastic orderings to contingent claim price comparison under different hedging portfolio constraints are also provided. (W t ) t≥0 Standard Brownian motion (B t ) t≥0 G-Brownian motion

show abstract

“…A stochastic ordering via sublinear expectations has also been defined in [92] by combining (4.9) with the inequality…”

Section: Comparison Results For G-risk Measuresmentioning

confidence: 99%

Section: G-stochastic Orderingsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stochastic orderings by nonlinear expectations

Ly¹

View full text Add to dashboard Cite

show abstract

$ G $-expectation approach to stochastic ordering

Privault

2022

FMF

View full text Add to dashboard Cite

<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>

show abstract

Stochastic ordering by <i>g</i>-expectations

Ly¹,

Privault²

2021

PUQR

View full text Add to dashboard Cite

We derive sufficient conditions for the convex and monotonic gstochastic ordering of diffusion processes under nonlinear g-expectations and gevaluations. Our approach relies on comparison results for forward-backward stochastic differential equations and on several extensions of convexity, monotonicity, and continuous dependence properties for the solutions of associated semilinear parabolic partial differential equations. Applications to contingent claim price comparison under different hedging portfolio constraints are provided.

show abstract

Uncertainty orders on the sublinear expectation space

Cited by 3 publications

References 21 publications

Stochastic orderings by nonlinear expectations

Stochastic orderings by nonlinear expectations

$ G $-expectation approach to stochastic ordering

Stochastic ordering by <i>g</i>-expectations

Contact Info

Product

Resources

About