Journal•ISSN: 0168-9673
Acta Mathematicae Applicatae Sinica
Springer Science+Business Media
About: Acta Mathematicae Applicatae Sinica is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Nonlinear system & Estimator. It has an ISSN identifier of 0168-9673. Over the lifetime, 2908 publications have been published receiving 13885 citations. The journal is also known as: Acta mathematicae applicatae Sinica (Print) & Yingyong shuxue xuebao.
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TL;DR: In this paper, the authors introduced the geometric process which is a sequence of independent nonnegative random variables, such that the distribution function of a random variable X n is F (a n−1 −1 normalized x), wherea is a positive constant, and the explicit expressions of the long-run average costs per unit time under each replacement policy are calculated, and therefore the corresponding optimal replacement policies can be found analytically or numerically.
Abstract: In this paper, we introduce and study the geometric process which is a sequence of independent non-negative random variablesX
1,X
2,... such that the distribution function ofX
n isF (a
n−1
x), wherea is a positive constant. Ifa>1, then it is a decreasing geometric process, ifa<1, it is an increasing geometric process. Then, we consider a replacement model as follows: the successive survival times of the system after repair form a decreasing geometric process or a renewal process while the consecutive repair times of the system constitute an increasing geometric process or a renewal process. Besides the replacement policy based on the working age of the system, a new kind of replacement policy which is determined by the number of failures is considered. The explicit expressions of the long-run average costs per unit time under each replacement policy are then calculated, and therefore the corresponding optimal replacement policies can be found analytically or numerically.
266 citations
TL;DR: In this article, a weakly compact family of probability measures (Pθ: θ ∈ gJ) representing an important sublinear expectation, G-expectation, was introduced.
Abstract: We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ: θ ∈ gJ} representing an important sublinear expectation— G-expectation \( \mathbb{E} \)[·]. We also give a concrete approximation of a bounded continuous function X(ω) by an increasing sequence of cylinder functions Lip(Ω) in order to prove that Cb(Ω) belongs to the completion of Lip(Ω) under the natural norm \( \mathbb{E} \)[| · |].
177 citations
Journal Article•
TL;DR: In this article, an approach to solving Hamilton-Jacobi equations (HJEs,for short) with parameters is presented, and both infinite and finite horizon nonlinear stochastic H_∞designs are also developed.
Abstract: This paper discusses the H_∞control problem for a class of nonlinear stochastic systems driven by Brownian motion with both the control and exogenous disturbance entering the diffusion.By an inequality of quadric form,an approach to solving Hamilton-Jacobi equations(HJEs,for short)with parameters is presented,and both infinite and finite horizon nonlinear stochastic H_∞designs are also developed.Finally,we discuss the solvability of such HJEs with parameters.
142 citations
TL;DR: In this article, the authors studied the problem of super-evaluation in a nonlinear Markovian setting and proved that the super evaluation is a filtration consistent nonlinear expectation.
Abstract: We will study the following problem. Let X
t
, t ∈ [0, T], be an R
d
–valued process defined on a time interval t ∈ [0, T]. Let Y be a random value depending on the trajectory of X. Assume that, at each fixed time t ≤ T, the information available to an agent (an individual, a firm, or even a market) is the trajectory of X before t. Thus at time T , the random value of Y (ω) will become known to this agent. The question is: how will this agent evaluate Y at the time t? We will introduce an evaluation operator e
t
[Y ] to define the value of Y given by this agent at time t. This operator e
t
[·] assigns an (X
s
)0≤s≤T
–dependent random variable Y to an (X
s
)0≤s≤t
–dependent random variable e
t
[Y ]. We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) with the drift coefficient b and diffusion coefficient σ containing an unknown parameter θ = θ
t
. We then consider the so called super evaluation when the agent is a seller of the asset Y . We will prove that such super evaluation is a filtration consistent nonlinear expectation. In some typical situations, we will prove that a filtration consistent nonlinear evaluation dominated by this super evaluation is a g–evaluation. We also consider the corresponding nonlinear Markovian situation.
133 citations
TL;DR: In this paper, the homogenized operator for a general sequence of linear parabolic operators is constructed for a given sequence of differential operators, and a method of constructing the homogeneized operator is presented.
Abstract: In this note, we present a method of constructing the homogenized operator for a general sequence of differential operators. As an example, we construct the homogenized operator for a sequence of linear parabolic operators.
123 citations