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Modeling with Stochastic Differential Equations
Published in Sandip Banerjee, Mathematical Modeling, 2021
A Wiener process or a Brownian motion is a zero-mean continuous process with independent Gaussian increments (by independent increments we mean a process Xt, where for every sequence t0<t1<⋯<tn, the random variables Xt1−Xt0,Xt2−Xt1,⋯,Xtn−Xtn−1 are independent).
Probability, Random Variables, and Stochastic Processes
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
The change in the process X(t2)−Xt1,X(t3)−X(t2),X(t4)−X(t3),⋯ between successive sampling instants, is called the increments of the process. A process in which the increments are all independent is said to be an independent increments process. Similarly, a process with stationary increments is said to be a stationary increments process. The stationary properties of discrete time random processes are similarly defined.
Stochastic Processes and an Introduction to Stochastic Differential Equations
Published in Henry C. Tuckwell, Elementary Applications of Probability Theory, 2018
We have already encountered one example of an independent-increment process in section 9.2 – the Poisson process. Before defining a Wiener process, we mention that if the distributions of the increments of a process in various time intervals depend only on the lengths of those intervals and not their locations (i.e., their starting values), then the increments are said to be stationary. In section 9.2 we saw that for a Poisson process N = {N(t), t ⩾ 0}, the random increment N(t2) − N(t1) is Poisson distributed with a parameter proportional to the length of the interval (t1, t2]. Thus a Poisson process has stationary independent increments.
Saddle-point equilibrium for Hurwicz model considering zero-sum differential game of uncertain dynamical systems with jump
Published in International Journal of Systems Science, 2023
Xi Li, Qiankun Song, Yurong Liu, Fuad E. Alsaadi
An uncertain process is said to be the canonical Liu process, if the following conditions are satisfied: and nearly all the sample paths are Lipschitz continuous. has stationary and independent increments.Each increment is the normal uncertain variable with expected value 0 and variance , and its uncertain distribution satisfies
A novel mathematical optimization model for a preemptive multi-priority M/M/C queueing system of emergency department’s patients, a real case study in Iran
Published in IISE Transactions on Healthcare Systems Engineering, 2022
Erfaneh Ghanbari, Sogand Soghrati Ghasbe, Amir Aghsami, Fariborz Jolai
Triage is the process of determining the severity of a patient’s condition in the ED. Patients with the most severe conditions receive immediate treatment. Therefore, because many ED patients require prompt services, this study’s primary purpose is to identify this prioritization’s impact on the ED service system. In the spread of COVID-19 disease, our proposed structural framework assumes that patients arrive at the ED according to the Poisson process. The process of patients entering the system is a counting process in the form of {N(t), t ≥ 0}. Obviously, the number of events at the time of zero is zero, i.e., N (0) = 0. On the other hand, this process has the property of independent increments. Because the number of patients’ arrival in a certain period of time is independent of the number of arrivals in another period, that does not overlap. For example, the number of patients who arrive between 8 and 9 AM is completely independent of the number of patients who arrive between 10 and 11 AM. The process of arrival of patients also has the property of stationary increments. It means that the number of patients who enter the system at any interval of time depends only on the length of the interval. Because patients do not enter the system at any time and each patient may have problems entering the system at any time, the number of patients in the system depends only on the length of the interval and is independent of the time of occurrence. Given these three properties, it can be said that the Patients arrive at the emergency department according to a Poisson process.
Nash equilibrium and bang-bang property for the non-zero-sum differential game of multi-player uncertain systems with Hurwicz criterion
Published in International Journal of Systems Science, 2022
Xi Li, Qiankun Song, Yurong Liu, Fuad E. Alsaadi
An uncertain process is called a canonical Liu process, if it satisfies the following three conditions: and almost all the sample paths are Lipschitz continuous. has stationary and independent increments.Every increment is a normal uncertain variable with expected value 0 and variance , and its uncertain distribution satisfies