Sublinear Expectation Research Articles

Weak laws of large numbers for weighted sums of arrays of random variables under sublinear expectation

. In this article, we introduce a new concept of integrability called extended residually h-integrability with exponent r, which deals with weighted sums of arrays under sublinear expectation. The classical copy of this concept extends the residually h-integrability given by Wang, Hu, and Volodin (2018). On this basis, we obtain three forms of weak laws of large numbers for weighted sums of arrays of rowwise negatively dependent or rowwise independent random variables, respectively, under sublinear expectation. As corollaries, we get some weak laws of large numbers for arrays of random variables with normalizing factor k n under sublinear expectation.

On the rate of convergence for an α-stable central limit theorem under sublinear expectation

On the rate of convergence for an α-stable central limit theorem under sublinear expectation

Bayesian Lower and Upper Estimates for Ether Option Prices with Conditional Heteroscedasticity and Model Uncertainty

This paper aims to leverage Bayesian nonlinear expectations to construct Bayesian lower and upper estimates for prices of Ether options, that is, options written on Ethereum, with conditional heteroscedasticity and model uncertainty. Specifically, a discrete-time generalized conditional autoregressive heteroscedastic (GARCH) model is used to incorporate conditional heteroscedasticity in the logarithmic returns of Ethereum, and Bayesian nonlinear expectations are adopted to introduce model uncertainty, or ambiguity, about the conditional mean and volatility of the logarithmic returns of Ethereum. Extended Girsanov’s principle is employed to change probability measures for introducing a family of alternative GARCH models and their risk-neutral counterparts. The Bayesian credible intervals for “uncertain” drift and volatility parameters obtained from conjugate priors and residuals obtained from the estimated GARCH model are used to construct Bayesian superlinear and sublinear expectations giving the Bayesian lower and upper estimates for the price of an Ether option, respectively. Empirical and simulation studies are provided using real data on Ethereum in AUD. Comparisons with a model incorporating conditional heteroscedasticity only and a model capturing ambiguity only are presented.

Almost Sure Convergence of Weighted Sums for m-END Sequences under Sub-linear Expectations

Almost Sure Convergence of Weighted Sums for m-END Sequences under Sub-linear Expectations

The convergence properties for randomly weighted sums of widely negative dependent random variables under sub-linear expectations with related statistical applications

In this paper, we study the complete convergence and complete moment convergence for randomly weighted sums of arrays of rowwise widely negative dependent random variables in sub-linear expectation space under some appropriate conditions, which extend and improve the corresponding ones in classical probability space to the case of sub-linear expectation space. And we obtain a strong law of large numbers for the randomly weighted sums of arrays of rowwise widely negative dependent random variables. As applications of our main results, we not only present a result on the complete consistency for the weighted estimator in a nonparametric regression model, but also obtain the complete consistency for the least squares estimators in errors-in-variables regression models based on widely negative dependent errors under sub-linear expectations. We perform some numerical simulations to verify the validity of the theoretical results and a real example is analysed for illustration.

Strong Convergence Properties for Weighted Sums of Extended Negatively Dependent Random Variables Under Sub-linear Expectations with Statistical Applications

Strong Convergence Properties for Weighted Sums of Extended Negatively Dependent Random Variables Under Sub-linear Expectations with Statistical Applications

On complete convergence for weighted sums of m-widely acceptable random variables under sub-linear expectations and its statistical applications

. In this article, by means of Rosenthal-type inequality, we study the complete convergence and strong law of large numbers for weighted sums of m-widely accpetable random variables under sub-linear expectations, which extend some existing results in classical probability space. As applications, we investigate the complete consistency and strong consistency of weighted estimators in nonparametric regression models under sub-linear expectations. In addition, we provide some simulation studies to verify the finite sample performance of the theoretical results.

A convergence theorem for Crandall–Lions viscosity solutions to path-dependent Hamilton–Jacobi–Bellman PDEs

We establish a convergence theorem for Crandall–Lions viscosity solutions to path-dependent Hamilton–Jacobi–Bellman PDEs. Our proof is based on a novel convergence theorem for dynamic sublinear expectations and the stochastic representation of viscosity solutions as value functions.

A general law of the iterated logarithm for non-additive probabilities

Motivated by some interesting problems in mathematical economics, quantum mechanics and finance, non-additive probabilities have been used to describe the phenomena which are generally non-additive. In this paper, we further study the law of the iterated logarithm (LIL) for non-additive probabilities, based on existing results. Under the framework of sublinear expectation initiated by Peng, we give two convergence results of Vn≔∑i=1nXinϕ(n) under some reasonable assumptions, where {Xi}i=1∞ is a sequence of random variables and ϕ is a positive nondecreasing function. From these, a general LIL for non-additive probabilities is proved for negatively dependent and non-identically distributed random variables. It turns out that our result is a natural extension of the Kolmogorov LIL and the Hartman–Wintner LIL. Theorem 1 and Theorem 2 in this paper can be seen an extension of Theorem 1 in Chen and Hu (2014).

A note on the G-Itô Formula and a comment on “Averaging Principle for SDEs of Neutral Type Driven by G-Brownian Motion”

This is a comment on the paper Stoch. Dyn. 19(1) (2019) 1950004, doi:10.1142/S0219493719500047) titled “Averaging Principle for SDEs of Neutral Type Driven by G-Brownian Motion”, preceded by a note on the G-Itô formula. In the commented paper, the authors provided a wrong result and incorrect computations in their Theorem 4.1 and Appendix B, which probably result from the misapplication of the G-Itô Formula in the sublinear expectation theory. Hence, we provide a differential form of the G-Itô formula with a calculation table and the correct computation on the solution related to Theorem 4.1 of the commented paper.

Complete [formula omitted]th moment convergence of moving average processes for [formula omitted]-widely acceptable random variables under sub-linear expectations

In this article, we studied complete qth moment convergence of the moving average processes produced by m-widely acceptable (m-WA) random variables under sub-linear expectations. The results here extend those of the moving average processes generated by m-WOD random variables in probability.

Complete convergence theorems for weighted sums of extended negatively dependent random variables under sub-linear expectations

. This article aims to study and establish the Lai law for the weighted sum of extended negatively dependent random variables under sub-linear expectations. Using inequalities under sub-linear expectation spaces, we extend the complete convergence of weighted sums from classical probability space to sub-linear expectation space.

Strong Limit Theorems for Weighted Sums under the Sub-linear Expectations

Strong Limit Theorems for Weighted Sums under the Sub-linear Expectations

Stochastic averaging principle for neutral stochastic functional differential equations driven by G-Lévy process

In this paper, we aim to establish an averaging principle for neutral stochastic functional differential equations driven by G-Lévy processes with non-Lipschitz coefficients. Utilizing the theory of sub-linear expectations, we show that the solutions of the considered stochastic systems can be approximated by the solutions of averaged neutral stochastic functional differential equations driven by G-Lévy processes in the mean square convergence and convergence in the sense of capacity. Finally, we give an example to support our main results.

A monotone scheme for G-equations with application to the explicit convergence rate of robust central limit theorem

We propose a monotone approximation scheme for a class of fully nonlinear PDEs called G-equations. Such equations arise often in the characterization of G-distributed random variables in a sublinear expectation space. The proposed scheme is constructed recursively based on a piecewise constant approximation of the viscosity solution to the G-equation. We establish the convergence of the scheme and determine the convergence rate with an explicit error bound, using the comparison principles for both the scheme and the equation together with a mollification procedure. The first application is obtaining the convergence rate of Peng's robust central limit theorem with an explicit bound of Berry-Esseen type. The second application is an approximation scheme with its convergence rate for the Black-Scholes-Barenblatt equation.

Complete convergence for weighted sums of widely negative orthant dependent random variables under the sub-linear expectations

. In this article, we study and establish complete convergence for widely negative orthant dependent random variables under the sub-linear expectations, which extend some complete convergence theorems for widely negative orthant dependent random variables from the classical probability space to sub-linear expectation space.

Limit theorems for delayed sums under sublinear expectation

In this paper, we study analogues of the strong laws of large numbers for delayed sums under sublinear expectation. The necessary and sufficient conditions for such analogues of the strong laws of large numbers for delayed sums are established. As a corollary, the complete convergence theorem under sublinear expectation could be deduced. The results extend the classical ones in [1].

On the complete moment convergence of moving average processes generated by negatively dependent random variables under sub-linear expectations

<abstract><p>The moving average processes $ X_k = \sum_{i = -\infty}^{\infty}a_{i+k}Y_{i} $ are studied, where $ \{Y_i, -\infty < i < \infty\} $ is a double infinite sequence of negatively dependent random variables under sub-linear expectations, and $ \{a_i, -\infty < i < \infty\} $ is an absolutely summable sequence of real numbers. We establish the complete moment convergence of a moving average process under proper conditions, extending the corresponding results in classic probability space to those in sub-linear expectation space.</p></abstract>

Almost sure convergence for a class of dependent random variables under sub-linear expectations

<abstract><p>This article aimed to investigate the almost sure convergence theorem of widely negative orthant dependent (WNOD) random variables under sub-linear expectation space. The conclusions in this essay are an extension of the corresponding conclusions in the classical probability space.</p></abstract>

Strong law of large numbers for weighted sums of $ m $-widely acceptable random variables under sub-linear expectation space

<p>In this article, using the Fuk-Nagaev type inequality, we studied general strong law of large numbers for weighted sums of $ m $-widely acceptable ($ m $-WA, for short) random variables under sublinear expectation space with the integral condition</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \hat{\mathbb{E}} \left ( f^-\left ( \left | X \right | \right ) \right ) \le \mathrm{C}_\mathbb{V}\left ( f^-\left ( \left | X \right | \right ) \right )< \infty $\end{document} </tex-math></disp-formula></p><p>and $ Choquet $ integrals existence, respectively, where</p><p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ f\left ( x \right ) = x^{1/\beta }L\left ( x \right ) $\end{document} </tex-math></disp-formula></p><p>for $ \beta > 1 $, $ L\left (x \right) > 0 $ $ \left(x > 0\right) $ was a monotonic nondecreasing slowly varying function, and $ f^-\left (x \right) $ was the inverse function of $ f\left(x\right) $. One of the results included the Kolmogorov-type strong law of large numbers and the partial Marcinkiewicz-type strong law of large numbers for $ m $-WA random variables under sublinear expectation space. Besides, we obtained almost surely convergence for weighted sums of $ m $-WA random variables under sublinear expectation space. These results improved the corresponding results of Ma and Wu under sublinear expectation space.</p>