Multiple G-It\^{o} integral in the G-expectation space

In this paper, motivated by mathematic finance we introduce the multiple G-It\^{o} integral in the G-expectation space, then investigate how to calculate. We get the the relationship between Hermite polynomials and multiple G-It\^{o} integrals which is a natural extension of the classical result obtained by It\^{o} in 1951.Comment: 9 page

An Invariance Principle of G-Brownian Motion for the Law of the Iterated Logarithm under G-expectation

The classical law of the iterated logarithm (LIL for short)as fundamental limit theorems in probability theory play an important role in the development of probability theory and its applications. Strassen (1964) extended LIL to large classes of functional random variables, it is well known as the invariance principle for LIL which provide an extremely powerful tool in probability and statistical inference. But recently many phenomena show that the linearity of probability is a limit for applications, for example in finance, statistics. As while a nonlinear expectation--- G-expectation has attracted extensive attentions of mathematicians and economists, more and more people began to study the nature of the G-expectation space. A natural question is: Can the classical invariance principle for LIL be generalized under G-expectation space? This paper gives a positive answer. We present the invariance principle of G-Brownian motion for the law of the iterated logarithm under G-expectation

On the Conjecture of Berry Regarding a Bernoulli Two-Armed Bandit

In this paper, we study an independent Bernoulli two-armed bandit with unknown parameters ρ and λ, where ρ and λ have a pair of priori distributions such that dR(ρ)=CRρr0(1−ρ)r0′dμ(ρ),dL(λ)=CLλl0(1−λ)l0′dμ(λ) and μ is an arbitrary positive measure on [0,1]. Berry proposed the conjecture that, given a pair of priori distributions (R,L) of parameters ρ and λ, the arm with R is the current optimal choice if r0+r0′l0+l0′ and the expectation of ρ is not less than that of λ. We give an easily verifiable equivalent form of Berry’s conjecture and use it to prove that Berry’s conjecture holds when R and L are two-point distributions as well as when R and L are beta distributions and the number of trials N≤r0r0′+1

Ergodicity of invariant capacities

In this paper, we investigate capacity preserving transformations and their ergodicity. We obtain some limit properties under capacity spaces and then give the concept of ergodicity for a capacity preserving transformation. Based on this definition, we give several characterizations of ergodicity. In particular, we obtain a type of Birkhoff’s ergodic theorem and prove that the ergodicity of a transformation with respect to an upper probability is equivalent to a type of strong law of large numbers

First-Principles-Based Optimized Design of Fluoride Electrolytes for Sodium-Ion Batteries

Because of the abundance and low cost of sodium, sodium-ion batteries (SIBs) are next-generation energy storage mediums. Furthermore, SIBs have become an alternative option for large-scale energy storage systems. Because the electrolyte is a critical component of SIBs, fluorination is performed to improve the cycling performance of electrolytes. Based on the first-principles study, we investigated the effects of the type, quantity, and relative position relationships of three fluorinated units, namely -CF1, -CF2, and -CF3, on the cyclic ester molecule ethylene carbonate (EC) and the linear ether molecule 1,2-dimethoxylethane (DME). The optimal fluorination was proposed for EC and DME by studying the bond length, highest occupied molecular orbital, lowest unoccupied lowest orbital, and other relevant parameters. The results revealed that for EC, the optimal fluorination is 4 F fluorination based on four -CF1 units; for DME, CF3CF1CF1-, CF3CF2CF2-, CF3CF1CF2CF3, and CF3CF2CF2CF3, four combinations of three -CF1, -CF2, and -CF3 units are optimal. The designed fluorinated EC and DME exhibited a wide electrochemical stability window and high ionic solvation ability, which overcomes the drawback of conventional solvents and can improve SIB cycling performance