Invariant and ergodic nonlinear expectations for $G$-diffusion
        processes

Hu, Mingshang; Li, Hanwu; Wang, Falei; Zheng, Guoqiang

doi:10.1214/ecp.v20-3886

Cited by 11 publications

(13 citation statements)

References 17 publications

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“…The invariant sublinear expectation has not been studied much in the literature. As far as we know, so far there is only one work (Hu et al [27]) on the existence of invariant sublinear expectation for G-diffusion processes if the system is sufficiently dissipative.…”

Section: Dynamical Systems On Sublinear Expectation Spaces and Ergodicitymentioning

confidence: 99%

Ergodicity of Sublinear Markovian Semigroups

Feng¹,

Zhao²

2021

SIAM J. Math. Anal.

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In this paper, we study the ergodicity of an invariant sublinear expectation of sublinear Markovian semigroups. For this, we first develop an ergodic theory of an expectation preserving map on a sublinear expectation space. Ergodicity is defined as any invariant set in which either itself or its complement has 0 capacity. We prove, under a general sublinear expectation space setting, the equivalent relation between ergodicity and the corresponding transformation operator having simple eigenvalue 1, and also with Birkhoff-type strong law of large numbers if the sublinear expectation is regular. For a sublinear Markov process, we prove that its ergodicity is equivalent to the Markovian semigroup having eigenvalue 1, and the eigenvalue is simple in the space of bounded measurable functions. As an example we show that G-Brownian motion \{ Bt\} t\geq 0 on the unit circle has an invariant expectation and is ergodic if and only if \BbbE ( - (B 1 ) 2 ) < 0. Moreover, it is also proved in this case that the invariant expectation is regular and the canonical stationary process has no mean-uncertainty under the invariant expectation.

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Section: Dynamical Systems On Sublinear Expectation Spaces and Ergodicitymentioning

confidence: 99%

Ergodicity of Sublinear Markovian Semigroups

Feng¹,

Zhao²

2021

SIAM J. Math. Anal.

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“…基于非线性框架下的倒向随机微分方程理论, 文献 [57,63] 研究了非线性期望框架下的遍历倒向随机微分方程理论; Hu 和 Ji [55] 研究了 G-期望框架下的随机递归最优控制问题, 他们引入了一种全新的 "隐式分解" 方法, 证明了动态规划原理. 最近, 文献 [ 参考文献…”

Section: 非线性期望下的随机分析的进展unclassified

Theory, methods and meaning of nonlinear expectation theory

ShiGe¹

2017

Sci. Sin.-Math.

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This is a survey on the research developments of nonlinear expectation theory. We first recall the basic definition of a space of nonlinear expectation, and then, through the representation theorem and some examples of nonlinear i.i.d. (independent and identically distributed), to explain why this new framework can be applied to calculate and quantitatively analyze probabilistic and distributional uncertainty hidden behind a real world (possibly high-dimensional) data sequence. Then we introduce two fundamentally important nonlinear normal distribution and maxima distribution and the corresponding nonlinear law of large numbers and nonlinear central limit theorem, which are crucial and fundamental breakthroughs of this new research domain. A typical application is a basic algorithm named "φ-max-mean". We also present a basic continuous-time stochastic process-nonlinear Brownian motion and its stochastic calculus, including stochastic integral, stochastic differential equations, and the corresponding nonlinear martingale theory. This new theoretical framework has generalized the axiomatical probability theory founded by Kolmogorov (1933). The key difference is the notion of nonlinear expectationÊ, whose special linear case corresponds a probability space (Ω, F, P). It is the nonlinearity that allows us to quantitatively measure the uncertainty of probabilities and probabilistic distributions inhabited in our real world. Keywords nonlinear expectation, nonlinear normal distribution, nonlinear i.i.d., nonlinear large numbers theorem and central limit theorem, nonlinear Brownian motion and its stochastic calculus, nonlinear martingale theory, nonlinear Monté-Carlo method, φ-max-mean algorithm MSC(

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“…Therefore, the ergodic theory of this paper can provide new insight to the study of capacity. On the other hand, there are also other papers attempting to investigate the ergodicity in capacity spaces or sublinear expectation spaces from different angles, one can see [10] and [19] and the references therein. None of these papers dealt with dynamical property of processes especially the non-decomposable property.…”

Section: Introductionmentioning

confidence: 99%