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Random Vectors and Random Functions
Published in Richard M. Golden, Statistical Machine Learning, 2020
A counting measure is a countably additive measure, ν:F→[0,∞], on the measurable space (Ω,F) defined such that for each F∈F: (1) ν(F) is the number of elements in F when F is a finite set, and (2) ν(F) = +∞ when F is not a finite set.
Measure and Integration
Published in James K. Peterson, Basic Analysis IV: Measure Theory and Integration, 2020
Exercise 9.1.4Let X be the counting numbers, N, and S = P(N). Define μ, by μ(E) is the cardinality of E if E is a finite set and oc otherwise. Prove μ is a measure. This measure is called thecounting measure. Note that N = ∪n {1, … , n} for all n and μ({1,…, n}) = n, which implies μ is a σ-finite measure.
Pseudo almost automorphy of stochastic neutral partial functional differential equations with Lévy noise
Published in Applicable Analysis, 2023
Zhinan Xia, Qianlian Wu, Jinliang Chai
Given a Lévy process L, we define the process of jumps of L by , , where . For any Borel set B in , define the random counting measure where ♯ means the counting and is the indicator function. We write and called it the intensity measure associated with L. If a Borel set B in is bounded below (that is, , where is the closure of B), then almost surely for all and is a Poisson process with the intensity . So N is called Poisson random measure. For each and B bounded below, the compensated Poisson random measure is defined by .
Dynamical behaviour of stochastic competition system with harvest and Lévy jumps
Published in Journal of Control and Decision, 2020
Yan Liu, Yunzhou Zhang, Shangdong Zhu, Lei Wang, Le Zhou
In reality, the system may also suffer sudden environmental factors effecting and fluctuating, such as earthquakes, floods, toxic pollutants, epidemics and other sudden factors, which cannot be presented by stochastic system (1) and (2). Some researchers have showed that Lévy jumps could describe sudden random environmental perturbations (see Liu & Bai, 2016; Liu & Zhu, 2018; Wu, Zou, Wang, & Liu, 2014; Zhang, Tong, & Bao, 2014; Zhao & Yuan, 2016). Therefore it is interesting and significant to introduce Lévy jumps into the underlying population system for explaining these discontinuous jumps phenomena, which can provide a reasonable and more realistic model, which is also the motivation to study this issue secondly. From the viewpoint of biomathematics, Lévy jumps are discontinuous, including jumps at time . The jump time is always random, and the waiting time of jumps is dependent. We establish the stochastic competition system with harvest and Lévy jumps as follows: where denotes the left limit of , N is a Poisson counting measure with characteristic measure λ on a measurable subset Y of with and . Furthermore, we suppose that B and N are independent.
One kind of linear-quadratic zero-sum stochastic differential game with jumps
Published in International Journal of Control, 2022
Zhiyong Yu, Baokai Zhang, Feng Zhang
Let T>0 be a fixed finite time horizon and a filtered probability space satisfying the usual conditions. Denote by the expectation under and the conditional expectation given . Let be a measure space with . Let be an adapted stationary Poisson point process with characteristic measure ν, where is a countable subset of . The counting measure induced by η is It's known that is the compensated Poisson random measure. Let be a d-dimensional standard Brownian motion which is independent of the random measure . Without loss of generality, is assumed to be the -augmentation of the natural filtration generated by the Brownian motion and the Poisson random measure. Denote by the -predictable σ-field on .