The martingale comparison method for Markov processes

2021 ◽  
Vol 58 (1) ◽  
pp. 164-176
Author(s):  
Benedikt Köpfer ◽  
Ludger Rüschendorf
Keyword(s):  
Markov Processes ◽  
Martingale Problem ◽  
Function Class ◽  
Comparison Method ◽  
Stochastic Orders ◽  
Regularity Conditions ◽  
Evolution System ◽  
Basic Step ◽  
Theory Of Evolution ◽  
Comparison Results

AbstractComparison results for Markov processes with respect to function-class-induced (integral) stochastic orders have a long history. The most general results so far for this problem have been obtained based on the theory of evolution systems on Banach spaces. In this paper we transfer the martingale comparison method, known for the comparison of semimartingales to Markovian semimartingales, to general Markov processes. The basic step of this martingale approach is the derivation of the supermartingale property of the linking process, giving a link between the processes to be compared. This property is achieved using the characterization of Markov processes by the associated martingale problem in an essential way. As a result, the martingale comparison method gives a comparison result for Markov processes under a general alternative but related set of regularity conditions compared to the evolution system approach.

scholarly journals Comparison Results for GARCH Processes

2014 ◽  
Vol 51 (3) ◽  
pp. 685-698
Author(s):  
Fabio Bellini ◽  
Franco Pellerey ◽  
Carlo Sgarra ◽  
Salimeh Yasaei Sekeh
Keyword(s):  
Stochastic Orders ◽  
Stochastic Comparison ◽  
Convex Order ◽  
Comparison Results ◽  
Garch Processes ◽  
Garch Process

We consider the problem of stochastic comparison of general GARCH-like processes for different parameters and different distributions of the innovations. We identify several stochastic orders that are propagated from the innovations to the GARCH process itself, and we discuss their interpretations. We focus on the convex order and show that in the case of symmetric innovations it is also propagated to the cumulated sums of the GARCH process. More generally, we discuss multivariate comparison results related to the multivariate convex and supermodular orders. Finally, we discuss ordering with respect to the parameters in the GARCH(1, 1) case.

scholarly journals A unified approach for drawdown (drawup) of time-homogeneous Markov processes

2017 ◽  
Vol 54 (2) ◽  
pp. 603-626 ◽  
Author(s):  
David Landriault ◽  
Bin Li ◽  
Hongzhong Zhang
Keyword(s):  
Markov Processes ◽  
Financial Risk ◽  
First Passage Time ◽  
Passage Time ◽  
Regularity Conditions ◽  
Unified Approach ◽  
First Passage ◽  
Underlying Process ◽  
Short Time ◽  
General Time

AbstractDrawdown (respectively, drawup) of a stochastic process, also referred as the reflected process at its supremum (respectively, infimum), has wide applications in many areas including financial risk management, actuarial mathematics, and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the drawdown (respectively, drawup) process, its overshoot, and the maximum of the underlying process at this first passage time. By using short-time pathwise analysis, under some mild regularity conditions, the joint law of the three drawdown quantities is shown to be the unique solution to an integral equation which is expressed in terms of fundamental two-sided exit quantities of the underlying process. Explicit forms for this joint law are found when the Markov process has only one-sided jumps or is a Lévy process (possibly with two-sided jumps). The proposed methodology provides a unified approach to study various drawdown quantities for the general class of time-homogeneous Markov processes.

Motion Comparison Method Combining Segmented Multi-Joint Line Graphs with the SIFT Matching Technique

2014 ◽  
Vol 599-601 ◽  
pp. 1566-1570
Author(s):  
Ming Zeng ◽  
Hong Lin Ren ◽  
Qing Hao Meng ◽  
Chang Wei Chen ◽  
Shu Gen Ma
Keyword(s):  
Feature Matching ◽  
Line Graph ◽  
Comparison Method ◽  
Joint Line ◽  
Line Graphs ◽  
3D Motion ◽  
Motion Data ◽  
Comparison Results ◽  
Matching Technique ◽  
Sift Features

In this paper, an effective motion comparison method based on segmented multi-joint line graphs combined with the SIFT feature matching method is proposed. Firstly, the multi-joint 3D motion data are captured using the Kinect. Secondly, 3D motion data are normalized and distortion data are removed. Therefore, a 2D line graph can be obtained. Next, SIFT features of the 2D motion line graph are extracted. Finally, the line graphs are divided into several regions and then the comparison results can be calculated based on SIFT matching ratios between the tutor’s local line graph and the trainee’s local line graph. The experimental results show that the proposed method not only can easily deal with the several challenge problems in motion analysis, e.g., the problem of different rhythm of motions, the problem of a large amount of data, but also can provide detailed error correction cues.

scholarly journals Stochastic orderings of multivariate elliptical distributions

2021 ◽  
Vol 58 (2) ◽  
pp. 551-568
Author(s):  
Chuancun Yin
Keyword(s):  
Necessary Conditions ◽  
Stochastic Orders ◽  
Regularity Conditions ◽  
Elliptical Distributions ◽  
Multivariate Normal ◽  
Random Vectors ◽  
Stochastic Orderings ◽  
Sufficient And Necessary Conditions ◽  
New Inequalities ◽  
Unified Method

AbstractFor two n-dimensional elliptical random vectors X and Y, we establish an identity for $\mathbb{E}[f({\bf Y})]- \mathbb{E}[f({\bf X})]$, where $f\,{:}\, \mathbb{R}^n \rightarrow \mathbb{R}$ satisfies some regularity conditions. Using this identity we provide a unified method to derive sufficient and necessary conditions for classifying multivariate elliptical random vectors according to several main integral stochastic orders. As a consequence we obtain new inequalities by applying the method to multivariate elliptical distributions. The results generalize the corresponding ones for multivariate normal random vectors in the literature.

On the differential equations for the transition probabilities of Markov processes with enumerably many states

1953 ◽  
Vol 49 (2) ◽  
pp. 247-262 ◽  
Author(s):  
G. E. H. Reuter ◽  
W. Ledermann ◽  
M. S. Bartlett
Keyword(s):  
Markov Process ◽  
Differential Equations ◽  
Conditional Probability ◽  
Markov Processes ◽  
Transition Probabilities ◽  
Regularity Conditions ◽  
Kolmogorov Equations ◽  
Image Position ◽  
Positive Integers

Let pik (s, t) (i, k = 1, 2, …; s ≤ t) be the transition probabilities of a Markov process in a system with an enumerable set of states. The states are labelled by positive integers, and pik (s, t) is the conditional probability that the system be in state k at time t, given that it was in state i at an earlier time s. If certain regularity conditions are imposed on the pik, they can be shown to satisfy the well-known Kolmogorov equations§

scholarly journals Comparison Results for GARCH Processes

2014 ◽  
Vol 51 (03) ◽  
pp. 685-698
Author(s):  
Fabio Bellini ◽  
Franco Pellerey ◽  
Carlo Sgarra ◽  
Salimeh Yasaei Sekeh
Keyword(s):  
Stochastic Orders ◽  
Stochastic Comparison ◽  
Convex Order ◽  
Comparison Results ◽  
Garch Processes ◽  
Garch Process

We consider the problem of stochastic comparison of general GARCH-like processes for different parameters and different distributions of the innovations. We identify several stochastic orders that are propagated from the innovations to the GARCH process itself, and we discuss their interpretations. We focus on the convex order and show that in the case of symmetric innovations it is also propagated to the cumulated sums of the GARCH process. More generally, we discuss multivariate comparison results related to the multivariate convex and supermodular orders. Finally, we discuss ordering with respect to the parameters in the GARCH(1, 1) case.

scholarly journals Manufacturing applications of the VIKOR approach: A scoping review

2021 ◽  
Vol 9 (4) ◽  
pp. 1673-1695
Author(s):  
Neylan Kaya
Keyword(s):  
Decision Making ◽  
Relevant Literature ◽  
Method Comparison ◽  
Comparison Method ◽  
Journal Title ◽  
Multi Criteria Decision Making ◽  
Vikor Method ◽  
Weighting Method ◽  
Comparison Results ◽  
Sorting Technique

VIKOR is a selecting and sorting technique for addressing problems and optimising multi-criteria decision making in complex systems. This study sought the relevant literature to categorise, analyse, and discuss the content and extent of existing studies that used the VIKOR method for applications in manufacturing. The study examined 84 studies published throughout 2018–2020. The studies were categorised by publication dates, author(s) name, techniques and methods, weighting method, comparison method, description of comparison results (comparing a given method to others), testing applicability, and journal-title. Analyses revealed that approximately 35 of the published studies involving VIKOR were related to its strategic use in manufacturing decisions and applications. In 2019, manufacturing was represented more than any other field among all published VIKOR papers, and Sustainability published more of the VIKOR-related articles than any other journal. Interestingly, the integrated and fuzzy VIKOR methods were used more than the traditional VIKOR method. Furthermore, the review results show that VIKOR is flexible enough to be continuously improved by integrating it with the new multi-criteria decision-making methods. This literature review can guide researchers and practitioners in applying VIKOR in various manufacturing fields.      

scholarly journals DETEKSI DINI PENYAKIT BALITA MENGGUNAKAN ALGORITMA SORENSEN BERBOBOT

2021 ◽  
Vol 9 (02) ◽  
pp. 60-67
Author(s):  
Nur Kharisa Umami ◽  
Setyawan Wibisono
Keyword(s):  
Expert System ◽  
Pairwise Comparison ◽  
Comparison Method ◽  
Case Based Reasoning ◽  
Analytic Hierarchy ◽  
Comparison Results ◽  
Similarity Algorithm ◽  
Analytic Hierarchy Process Method ◽  
Hierarchy Process ◽  
Case Based

There are still many parents who do not have sufficient understanding in terms of toddler disease. One way to provide education is the availability of a system that can be used for consultation based on the symptoms of illness experienced by toddlers and the actions needed to overcome them. The system that will be built is an expert system that can relatively provide suggestions for solutions to children's health problems using the Case Based Reasoning (CBR) method. namely an expert system that uses case-based reasoning methods, namely looking for similarities of a disease compared to a disease that has existed before. In this study, the CBR method was combined with a weighting process using the pairwise comparison method which was within the scope of the AHP (Analytic Hierarchy Process) method. In comparing consultations with old diseases that already exist in the system, and looking for similarities from the comparison results, the Sorensen similarity algorithm is used. This study resulted in weights with 3 symptom categories, namely mild symptoms with a weight of 0.09, moderate symptoms with a weight of 0.24 and severe symptoms with a weight of 0.67 and will recommend several diseases with a similarity above 0.5 and diseases with a similarity below 0.5 will be entered into the revise table to find a solution.

Nonlinear filtering of stochastic dynamical systems with Lévy noises

2015 ◽  
Vol 47 (03) ◽  
pp. 902-918 ◽  
Author(s):  
Huijie Qiao ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Filtering ◽  
Martingale Problem ◽  
Strong Solutions ◽  
Observation System ◽  
Regularity Conditions ◽  
Signal System ◽  
Zakai Equation ◽  
Stochastic Dynamical Systems ◽  
Non Gaussian

Nonlinear filtering is investigated in a system where both the signal system and the observation system are under non-Gaussian Lévy fluctuations. Firstly, the Zakai equation is derived, and it is further used to derive the Kushner-Stratonovich equation. Secondly, by a filtered martingale problem, uniqueness for strong solutions of the Kushner-Stratonovich equation and the Zakai equation is proved. Thirdly, under some extra regularity conditions, the Zakai equation for the unnormalized density is also derived in the case of α-stable Lévy noise.

scholarly journals Level-crossing ordering of semi-Markov processes and markov chains

2005 ◽  
Vol 42 (04) ◽  
pp. 989-1002 ◽  
Author(s):  
Fátima Ferreira ◽  
António Pacheco
Keyword(s):  
Markov Chains ◽  
Markov Processes ◽  
Stochastic Order ◽  
Sample Path ◽  
Stochastic Orders ◽  
Level Crossing ◽  
The Times ◽  
Definition Of ◽  
The Right ◽  
Semi Markov Processes

We extend the definition of level-crossing ordering of stochastic processes, proposed by Irle and Gani (2001), to the case in which the times to exceed levels are compared using an arbitrary stochastic order, and work, in particular, with integral stochastic orders closed for convolution. Using a sample-path approach, we establish level-crossing ordering results for the case in which the slower of the processes involved in the comparison is skip-free to the right. These results are specially useful in simulating processes that are ordered in level crossing, and extend results of Irle and Gani (2001), Irle (2003), and Ferreira and Pacheco (2005) for skip-free-to-the-right discrete-time Markov chains, semi-Markov processes, and continuous-time Markov chains, respectively.

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