Weighted exponential inequality for differentially subordinate martingales

The paper contains a study of weighted exponential inequalities for differentially subordinate martingales, under the assumption that the underlying weight satisfies Muckenhoupt’s condition $$A_{\infty }$$ A ∞ . The proof exploits certain functions enjoying appropriate size conditions and concavity. The martingales are adapted, uniformly integrable, and càdlàg - we do not assume any path-continuity restrictions. Because of this generality, we need to handle jump parts of processes which forces us to construct a Bellman function satisfying a stronger condition than local concavity. As a corollary, we will establish some new weighted $$L^p$$ L p estimates for differential subordinates of bounded martingales.

Relaxation schemes for the joint linear chance constraint based on probability inequalities

<p style='text-indent:20px;'>This paper is concerned with the joint chance constraint for a system of linear inequalities. We discuss computationally tractble relaxations of this constraint based on various probability inequalities, including Chebyshev inequality, Petrov exponential inequalities, and others. Under the linear decision rule and additional assumptions about first and second order moments of the random vector, we establish several upper bounds for a single chance constraint. This approach is then extended to handle the joint linear constraint. It is shown that the relaxed constraints are second-order cone representable. Numerical test results are presented and the problem of how to choose proper probability inequalities is discussed.</p>

On the laws of the iterated logarithm under sub-linear expectations

<p style='text-indent:20px;'>In this paper, we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space, where the random variables are not necessarily identically distributed. Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm. As an application, the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained. In the paper, it is also shown that if the sub-linear expectation space is rich enough, it will have no continuous capacity. The laws of the iterated logarithm are established without the assumption on the continuity of capacities.</p>