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Journal articles on the topic 'Law Invariant Risk Measures'

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Published: 9 March 2023
Last updated: 10 March 2023

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1

Ekeland, Ivar, and Walter Schachermayer. "Law invariant risk measures onL∞(ℝd)." Statistics & Risk Modeling 28, no. 3 (September 2011): 195–225. http://dx.doi.org/10.1524/stnd.2011.1099.

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2

Cherny, Alexander S., and Pavel G. Grigoriev. "Dilatation monotone risk measures are law invariant." Finance and Stochastics 11, no. 2 (February 8, 2007): 291–98. http://dx.doi.org/10.1007/s00780-007-0034-8.

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3

Lacker, Daniel. "Law invariant risk measures and information divergences." Dependence Modeling 6, no. 1 (November 1, 2018): 228–58. http://dx.doi.org/10.1515/demo-2018-0014.

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AbstractAone-to-one correspondence is drawnbetween lawinvariant risk measures and divergences,which we define as functionals of pairs of probability measures on arbitrary standard Borel spaces satisfying a few natural properties. Divergences include many classical information divergence measures, such as relative entropy and convex f -divergences. Several properties of divergence and their duality with law invariant risk measures are characterized, such as joint semicontinuity and convexity, and we notably relate their chain rules or additivity properties with certain notions of time consistency for dynamic law risk measures known as acceptance and rejection consistency. The examples of shortfall risk measures and optimized certainty equivalents are discussed in detail.
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4

Chen, Shengzhong, Niushan Gao, and Foivos Xanthos. "The strong Fatou property of risk measures." Dependence Modeling 6, no. 1 (October 1, 2018): 183–96. http://dx.doi.org/10.1515/demo-2018-0012.

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AbstractIn this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property,whichwas introduced in [19], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on a rearrangement invariant space X with the strong Fatou property is (X, L1) lower semicontinuous and that the converse is true on a wide range of rearrangement invariant spaces. We also study inf-convolutions of law-invariant or surplus-invariant risk measures that preserve the (strong) Fatou property.
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5

Chen, Shengzhong, Niushan Gao, Denny H. Leung, and Lei Li. "Automatic Fatou property of law-invariant risk measures." Insurance: Mathematics and Economics 105 (July 2022): 41–53. http://dx.doi.org/10.1016/j.insmatheco.2022.03.007.

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6

Cheung, K. C., K. C. J. Sung, S. C. P. Yam, and S. P. Yung. "Optimal reinsurance under general law-invariant risk measures." Scandinavian Actuarial Journal 2014, no. 1 (December 23, 2011): 72–91. http://dx.doi.org/10.1080/03461238.2011.636880.

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7

Xin, Linwei, and Alexander Shapiro. "Bounds for nested law invariant coherent risk measures." Operations Research Letters 40, no. 6 (November 2012): 431–35. http://dx.doi.org/10.1016/j.orl.2012.09.002.

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8

Shapiro, Alexander. "On Kusuoka Representation of Law Invariant Risk Measures." Mathematics of Operations Research 38, no. 1 (February 2013): 142–52. http://dx.doi.org/10.1287/moor.1120.0563.

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9

CHEN, YANHONG, and YIJUN HU. "SET-VALUED LAW INVARIANT COHERENT AND CONVEX RISK MEASURES." International Journal of Theoretical and Applied Finance 22, no. 03 (May 2019): 1950004. http://dx.doi.org/10.1142/s0219024919500043.

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In this paper, we investigate representation results for set-valued law invariant coherent and convex risk measures, which can be considered as a set-valued extension of the multivariate scalar law invariant coherent and convex risk measures studied in the literature. We further introduce a new class of set-valued risk measures, named set-valued distortion risk measures, which can be considered as a set-valued version of multivariate scalar distortion risk measures introduced in the literature. The relationship between set-valued distortion risk measures and set-valued weighted value at risk is also given.
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10

Belomestny, Denis, and Volker Krätschmer. "Central Limit Theorems for Law-Invariant Coherent Risk Measures." Journal of Applied Probability 49, no. 1 (March 2012): 1–21. http://dx.doi.org/10.1239/jap/1331216831.

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In this paper we study the asymptotic properties of the canonical plugin estimates for law-invariant coherent risk measures. Under rather mild conditions not relying on the explicit representation of the risk measure under consideration, we first prove a central limit theorem for independent and identically distributed data, and then extend it to the case of weakly dependent data. Finally, a number of illustrating examples is presented.
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11

Belomestny, Denis, and Volker Krätschmer. "Central Limit Theorems for Law-Invariant Coherent Risk Measures." Journal of Applied Probability 49, no. 01 (March 2012): 1–21. http://dx.doi.org/10.1017/s0021900200008834.

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In this paper we study the asymptotic properties of the canonical plugin estimates for law-invariant coherent risk measures. Under rather mild conditions not relying on the explicit representation of the risk measure under consideration, we first prove a central limit theorem for independent and identically distributed data, and then extend it to the case of weakly dependent data. Finally, a number of illustrating examples is presented.
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12

Angelsberg, Gilles, Freddy Delbaen, Ivo Kaelin, Michael Kupper, and Joachim Näf. "On a class of law invariant convex risk measures." Finance and Stochastics 15, no. 2 (December 23, 2010): 343–63. http://dx.doi.org/10.1007/s00780-010-0145-5.

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13

Krätschmer, Volker, Alexander Schied, and Henryk Zähle. "Comparative and qualitative robustness for law-invariant risk measures." Finance and Stochastics 18, no. 2 (January 16, 2014): 271–95. http://dx.doi.org/10.1007/s00780-013-0225-4.

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14

Grechuk, Bogdan, Anton Molyboha, and Michael Zabarankin. "CHEBYSHEV INEQUALITIES WITH LAW-INVARIANT DEVIATION MEASURES." Probability in the Engineering and Informational Sciences 24, no. 1 (December 21, 2009): 145–70. http://dx.doi.org/10.1017/s0269964809990192.

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The consistency of law-invariant general deviation measures with concave ordering has been used to generalize the Rao–Blackwell theorem and to develop an approach for reducing minimization of law-invariant deviation measures to minimization of the measures on subsets of undominated random variables with respect to concave ordering. This approach has been applied for constructing the Chebyshev and Kolmogorov inequalities with law-invariant deviation measures—in particular with mean absolute deviation, lower semideviation and conditional value-at-risk deviation. Additionally, an advantage of the Kolmogorov inequality with certain deviation measures has been illustrated in estimating the probability of the exchange rate of two currencies to be within specified bounds.
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15

Choi, Sungyong, and Andrzej Ruszczyński. "A risk-averse newsvendor with law invariant coherent measures of risk." Operations Research Letters 36, no. 1 (January 2008): 77–82. http://dx.doi.org/10.1016/j.orl.2007.04.008.

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16

Choi, Sungyong, Andrzej Ruszczyński, and Yao Zhao. "A Multiproduct Risk-Averse Newsvendor with Law-Invariant Coherent Measures of Risk." Operations Research 59, no. 2 (April 2011): 346–64. http://dx.doi.org/10.1287/opre.1100.0896.

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17

FÖLLMER, HANS, and THOMAS KNISPEL. "ENTROPIC RISK MEASURES: COHERENCE VS. CONVEXITY, MODEL AMBIGUITY AND ROBUST LARGE DEVIATIONS." Stochastics and Dynamics 11, no. 02n03 (September 2011): 333–51. http://dx.doi.org/10.1142/s0219493711003334.

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We study a coherent version of the entropic risk measure, both in the law-invariant case and in a situation of model ambiguity. In particular, we discuss its behavior under the pooling of independent risks and its connection with a classical and a robust large deviations bound.
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18

Filipović, Damir, and Gregor Svindland. "THE CANONICAL MODEL SPACE FOR LAW-INVARIANT CONVEX RISK MEASURES IS L1." Mathematical Finance 22, no. 3 (June 7, 2012): 585–89. http://dx.doi.org/10.1111/j.1467-9965.2012.00534.x.

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19

Gao, Niushan, Denny Leung, Cosimo Munari, and Foivos Xanthos. "Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces." Finance and Stochastics 22, no. 2 (March 13, 2018): 395–415. http://dx.doi.org/10.1007/s00780-018-0357-7.

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20

Schied, Alexander. "On the Neyman–Pearson problem for law-invariant risk measures and robust utility functionals." Annals of Applied Probability 14, no. 3 (August 2004): 1398–423. http://dx.doi.org/10.1214/105051604000000341.

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21

Shapiro, Alexander. "Consistency of Sample Estimates of Risk Averse Stochastic Programs." Journal of Applied Probability 50, no. 2 (June 2013): 533–41. http://dx.doi.org/10.1239/jap/1371648959.

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In this paper we study asymptotic consistency of law invariant convex risk measures and the corresponding risk averse stochastic programming problems for independent, identically distributed data. Under mild regularity conditions, we prove a law of large numbers and epiconvergence of the corresponding statistical estimators. This can be applied in a straightforward way to establish convergence with probability 1 of sample-based estimators of risk averse stochastic programming problems.
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22

Shapiro, Alexander. "Consistency of Sample Estimates of Risk Averse Stochastic Programs." Journal of Applied Probability 50, no. 02 (June 2013): 533–41. http://dx.doi.org/10.1017/s0021900200013541.

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In this paper we study asymptotic consistency of law invariant convex risk measures and the corresponding risk averse stochastic programming problems for independent, identically distributed data. Under mild regularity conditions, we prove a law of large numbers and epiconvergence of the corresponding statistical estimators. This can be applied in a straightforward way to establish convergence with probability 1 of sample-based estimators of risk averse stochastic programming problems.
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23

Liu, Fangda, and Ruodu Wang. "A Theory for Measures of Tail Risk." Mathematics of Operations Research 46, no. 3 (August 2021): 1109–28. http://dx.doi.org/10.1287/moor.2020.1072.

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The notion of “tail risk” has been a crucial consideration in modern risk management and financial regulation, as very well documented in the recent regulatory documents. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures that quantify the tail risk, that is, the behaviour of a risk beyond a certain quantile. Such risk measures are referred to as tail risk measures in this paper. The two popular classes of regulatory risk measures in banking and insurance, value at risk (VaR) and expected shortfall, are prominent, yet elementary, examples of tail risk measures. We establish a connection between a tail risk measure and a corresponding law-invariant risk measure, called its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further relevant issues on tail risk measures, such as bounds, distortion risk measures, risk aggregation, elicitability, and dual representations. In particular, there is no elicitable tail convex risk measure other than the essential supremum, and under a continuity condition, the only elicitable and positively homogeneous monetary tail risk measures are the VaRs.
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24

Bielecki, Tomasz R., Igor Cialenco, Marcin Pitera, and Thorsten Schmidt. "Fair estimation of capital risk allocation." Statistics & Risk Modeling 37, no. 1-2 (March 1, 2020): 1–24. http://dx.doi.org/10.1515/strm-2019-0011.

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AbstractIn this paper, we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular focus on expected shortfall. We introduce the concept of fair capital allocations and provide explicit formulae for fair capital allocations in case when the constituents of the risky portfolio are jointly normally distributed. The main focus of the paper is on the problem of approximating fair portfolio allocations in the case of not fully known law of the portfolio constituents. We define and study the concepts of fair allocation estimators and asymptotically fair allocation estimators. A substantial part of our study is devoted to the problem of estimating fair risk allocations for expected shortfall. We study this problem under normality as well as in a nonparametric setup. We derive several estimators, and prove their fairness and/or asymptotic fairness. Last, but not least, we propose two backtesting methodologies that are oriented at assessing the performance of the allocation estimation procedure. The paper closes with a substantial numerical study of the subject and an application to market data.
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25

Chen, Mi, Wenyuan Wang, and Ruixing Ming. "Optimal Reinsurance Under General Law-Invariant Convex Risk Measure and TVaR Premium Principle." Risks 4, no. 4 (December 16, 2016): 50. http://dx.doi.org/10.3390/risks4040050.

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26

Li, Jonathan Yu-Meng. "Technical Note—Closed-Form Solutions for Worst-Case Law Invariant Risk Measures with Application to Robust Portfolio Optimization." Operations Research 66, no. 6 (November 2018): 1533–41. http://dx.doi.org/10.1287/opre.2018.1736.

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27

Wang, Ruodu, Yunran Wei, and Gordon E. Willmot. "Characterization, Robustness, and Aggregation of Signed Choquet Integrals." Mathematics of Operations Research 45, no. 3 (August 2020): 993–1015. http://dx.doi.org/10.1287/moor.2019.1020.

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This article contains various results on a class of nonmonotone, law-invariant risk functionals called the signed Choquet integrals. A functional characterization via comonotonic additivity is established along with some theoretical properties, including six equivalent conditions for a signed Choquet integral to be convex. We proceed to address two practical issues currently popular in risk management, namely robustness (continuity) issues and risk aggregation with dependence uncertainty, for signed Choquet integrals. Our results generalize in several directions those in the literature of risk functionals. From the results obtained in this paper, we see that many profound and elegant mathematical results in the theory of risk measures hold for the general class of signed Choquet integrals; thus, they do not rely on the assumption of monotonicity.
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28

Tadese, Mekonnen, and Samuel Drapeau. "Dual representation of expectile-based expected shortfall and its properties." Probability, Uncertainty and Quantitative Risk 6, no. 2 (2021): 99. http://dx.doi.org/10.3934/puqr.2021005.

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<p style='text-indent:20px;'>An expectile can be considered a generalization of a quantile. While expected shortfall is a quantile-based risk measure, we study its counterpart—the expectile-based expected shortfall—where expectile takes the place of a quantile. We provide its dual representation in terms of a Bochner integral. Among other properties, we show that it is bounded from below in terms of the convex combination of expected shortfalls, and also from above by the smallest law invariant, coherent, and comonotonic risk measures, for which we give the explicit formulation of the corresponding distortion function. As a benchmark to the industry standard expected shortfall, we further provide its comparative asymptotic behavior in terms of extreme value distributions. Based on these results, we finally explicitly compute the expectile-based expected shortfall for selected classes of distributions.</p>
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29

Gaigall, Daniel. "TEST FOR CHANGES IN THE MODELED SOLVENCY CAPITAL REQUIREMENT OF AN INTERNAL RISK MODEL." ASTIN Bulletin 51, no. 3 (August 6, 2021): 813–37. http://dx.doi.org/10.1017/asb.2021.20.

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AbstractIn the context of the Solvency II directive, the operation of an internal risk model is a possible way for risk assessment and for the determination of the solvency capital requirement of an insurance company in the European Union. A Monte Carlo procedure is customary to generate a model output. To be compliant with the directive, validation of the internal risk model is conducted on the basis of the model output. For this purpose, we suggest a new test for checking whether there is a significant change in the modeled solvency capital requirement. Asymptotic properties of the test statistic are investigated and a bootstrap approximation is justified. A simulation study investigates the performance of the test in the finite sample case and confirms the theoretical results. The internal risk model and the application of the test is illustrated in a simplified example. The method has more general usage for inference of a broad class of law-invariant and coherent risk measures on the basis of a paired sample.
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30

Ra, Kwang Hyun, and YeonSoo Kim. "Racialized perceptions of the police." Policing: An International Journal 42, no. 2 (April 8, 2019): 301–15. http://dx.doi.org/10.1108/pijpsm-11-2017-0144.

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PurposeThe purpose of this paper is to examine differences in latent structures/dimensions in public perceptions of the police by race/ethnicity and level of identification with a given race/ethnic group.Design/methodology/approachTo identify differences in dimensions of juveniles’ perceptions of the police by the sub-samples, factor analyses were conducted utilizing data from the Gang Resistance Education and Training program evaluation.FindingsThe results show that minority juveniles have a relatively fragmented dimensional structure for the construct of perceptions of the police, while white juveniles have a unidimensional structure. Furthermore, moderate within-group differences in structures were found among African–American juveniles.Research limitations/implicationsThe results of the current study call for further examination of racial invariant assumptions in criminology. Since individual dimensions constituting perceptions of the police vary by race/ethnicity, those dimensions may potentially have unique associations with endogenous variables (e.g. criminality and cooperation with the police) according to individuals’ racial/ethnic membership.Practical implicationsPolice should clearly understand individuals’ dimensions constituting perceptions of the police and should identify dimensions that greatly impact precursors to compliance and cooperation with police such as perceived police legitimacy or perceived risk of sanction.Originality/valueIndividuals’ dimensions constituting perceptions of the police have significant implications on the construction of measures and their associations with other variables; however, racial differences in these dimensions have not been explored since Sullivanet al.’s (1987) research about three decades ago. In addition, the current study examined within-race differences in the dimensions constituting perceptions of the police.
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31

Gao, Niushan, and Cosimo Munari. "Surplus-Invariant Risk Measures." Mathematics of Operations Research 45, no. 4 (November 2020): 1342–70. http://dx.doi.org/10.1287/moor.2019.1035.

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This paper presents a systematic study of the notion of surplus invariance, which plays a natural and important role in the theory of risk measures and capital requirements. So far, this notion has been investigated in the setting of some special spaces of random variables. In this paper, we develop a theory of surplus invariance in its natural framework, namely, that of vector lattices. Besides providing a unifying perspective on the existing literature, we establish a variety of new results including dual representations and extensions of surplus-invariant risk measures and structural results for surplus-invariant acceptance sets. We illustrate the power of the lattice approach by specifying our results to model spaces with a dominating probability, including Orlicz spaces, as well as to robust model spaces without a dominating probability, where the standard topological techniques and exhaustion arguments cannot be applied.
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32

Laczkovich, M. "Invariant signed measures and the cancellation law." Proceedings of the American Mathematical Society 111, no. 2 (February 1, 1991): 421. http://dx.doi.org/10.1090/s0002-9939-1991-1036988-2.

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33

Weber, Stefan. "Distribution-Invariant Risk Measures, Entropy, and Large Deviations." Journal of Applied Probability 44, no. 1 (March 2007): 16–40. http://dx.doi.org/10.1239/jap/1175267161.

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The simulation of distributions of financial assets is an important issue for financial institutions. If risk measures are evaluated for a simulated distribution instead of the model-implied distribution, the errors in the risk measurements need to be analyzed. For distribution-invariant risk measures which are continuous on compacts, we employ the theory of large deviations to study the probability of large errors. If the approximate risk measurements are based on the empirical distribution of independent samples, then the rate function equals the minimal relative entropy under a risk measure constraint. We solve this minimization problem explicitly for shortfall risk and average value at risk.
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34

Weber, Stefan. "Distribution-Invariant Risk Measures, Entropy, and Large Deviations." Journal of Applied Probability 44, no. 01 (March 2007): 16–40. http://dx.doi.org/10.1017/s0021900200002692.

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The simulation of distributions of financial assets is an important issue for financial institutions. If risk measures are evaluated for a simulated distribution instead of the model-implied distribution, the errors in the risk measurements need to be analyzed. For distribution-invariant risk measures which are continuous on compacts, we employ the theory of large deviations to study the probability of large errors. If the approximate risk measurements are based on the empirical distribution of independent samples, then the rate function equals the minimal relative entropy under a risk measure constraint. We solve this minimization problem explicitly for shortfall risk and average value at risk.
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35

Weber, Stefan. "DISTRIBUTION-INVARIANT RISK MEASURES, INFORMATION, AND DYNAMIC CONSISTENCY." Mathematical Finance 16, no. 2 (April 2006): 419–41. http://dx.doi.org/10.1111/j.1467-9965.2006.00277.x.

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36

Pflug, Georg, and Nancy Wozabal. "Asymptotic distribution of law-invariant risk functionals." Finance and Stochastics 14, no. 3 (January 21, 2010): 397–418. http://dx.doi.org/10.1007/s00780-009-0121-0.

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37

Stadje, Mitja. "Two results on dynamic extensions of deviation measures." Journal of Applied Probability 57, no. 2 (June 2020): 531–40. http://dx.doi.org/10.1017/jpr.2020.11.

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AbstractWe give a dynamic extension result of the (static) notion of a deviation measure. We also study distribution-invariant deviation measures and show that the only dynamic deviation measure which is law invariant and recursive is the variance.
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38

Tserpes, N. A. "A note on the support of right invariant measures." International Journal of Mathematics and Mathematical Sciences 15, no. 2 (1992): 405–8. http://dx.doi.org/10.1155/s016117129200053x.

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A regular measureμon a locally compact topological semigroup is called right invariant ifμ(Kx)=μ(K)for every compactKandxin its support. It is shown that this condition implies a property reminiscent of the right cancellation law. This is used to generalize a theorem of A. Mukherjea and the author (with a new proof) to the effect that the support of anr*-invariant measure is a left group iff the measure is right invariant on its support.
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39

Jouini, E., W. Schachermayer, and N. Touzi. "OPTIMAL RISK SHARING FOR LAW INVARIANT MONETARY UTILITY FUNCTIONS." Mathematical Finance 18, no. 2 (April 2008): 269–92. http://dx.doi.org/10.1111/j.1467-9965.2007.00332.x.

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40

Ernst, Dietmar. "Risk Measures in Simulation-Based Business Valuation: Classification of Risk Measures in Risk Axiom Systems and Application in Valuation Practice." Risks 11, no. 1 (January 6, 2023): 13. http://dx.doi.org/10.3390/risks11010013.

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Simulation-based company valuations are based on an analysis of the risks in the company to be valued. This means that risk analysis is decisively important in a simulation-based business valuation. The link between risk measures, risk conception and risk axiom systems has not yet been sufficiently elaborated for simulation-based business valuations. The aim of this study was to determine which understanding of risk underlies simulation-based business valuations and how this can be implemented via suitable risk measures in simulation-based business valuations. The contribution of this study is providing guidance for the methodologically correct selection of appropriate risk measures. This will help with avoiding valuation errors. To this end, the findings were combined from risk axiom systems with the valuation equations of simulation-based business valuations. Only position-invariant risk measures are suitable for simulation-based business valuations.
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41

Landsman, Z., and U. Makov. "Translation-invariant and positive-homogeneous risk measures and optimal portfolio management." European Journal of Finance 17, no. 4 (April 2011): 307–20. http://dx.doi.org/10.1080/1351847x.2010.481467.

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42

Baser, Raymond E., Yuelin Li, Debra Brennessel, M. Margaret Kemeny, and Jennifer L. Hay. "Measurement invariance of intuitive cancer risk perceptions across diverse populations: The Cognitive Causation and Negative Affect in Risk scales." Journal of Health Psychology 24, no. 9 (February 2017): 1221–32. http://dx.doi.org/10.1177/1359105317693910.

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Intuitive cancer risk perceptions may inform strategies to motivate cancer prevention behaviors. This study evaluated factor structure and measurement invariance of two new measures of intuitive cancer risk, the Cognitive Causation and Negative Affect in Risk scales. Single- and multiple-group confirmatory factor analysis models were fit to responses from three diverse samples. The confirmatory factor analysis models fit the data well, with all comparative fit indices (CFI) ≥ 0.94. Items flagged by chi-square difference tests as potentially non-invariant were largely invariant between samples according to practical fit indices (e.g. ΔCFI). These novel scales may be particularly relevant in diverse, underserved populations.
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43

OSIPENKO, GEORGE. "Symbolic images and invariant measures of dynamical systems." Ergodic Theory and Dynamical Systems 30, no. 4 (July 17, 2009): 1217–37. http://dx.doi.org/10.1017/s0143385709000431.

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AbstractLet f be a homeomorphism of a compact manifold M. The Krylov–Bogoloubov theorem guarantees the existence of a measure that is invariant with respect to f. The set of all invariant measures ℳ(f) is convex and compact in the weak topology. The goal of this paper is to construct the set ℳ(f). To obtain an approximation of ℳ(f), we use the symbolic image with respect to a partition C={M(1),M(2),…,M(n)} of M. A symbolic image G is a directed graph such that a vertex i corresponds to the cell M(i) and an edge i→j exists if and only if f(M(i))∩M(j)≠0̸. This approach lets us apply the coding of orbits and symbolic dynamics to arbitrary dynamical systems. A flow on the symbolic image is a probability distribution on the edges which satisfies Kirchhoff’s law at each vertex, i.e. the incoming flow equals the outgoing one. Such a distribution is an approximation to some invariant measure. The set of flows on the symbolic image G forms a convex polyhedron ℳ(G) which is an approximation to the set of invariant measures ℳ(f). By considering a sequence of subdivisions of the partitions, one gets sequence of symbolic images Gk and corresponding approximations ℳ(Gk) which tend to ℳ(f) as the diameter of the cells goes to zero. If the flows mk on each Gk are chosen in a special manner, then the sequence {mk} converges to some invariant measure. Every invariant measure can be obtained by this method. Applications and numerical examples are given.
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44

Fredes, Luis, and Jean-François Marckert. "Invariant measures of interacting particle systems: Algebraic aspects." ESAIM: Probability and Statistics 24 (2020): 526–80. http://dx.doi.org/10.1051/ps/2020008.

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Consider a continuous time particle system ηt = (ηt(k), k ∈ 𝕃), indexed by a lattice 𝕃 which will be either ℤ, ℤ∕nℤ, a segment {1, ⋯ , n}, or ℤd, and taking its values in the set Eκ𝕃 where Eκ = {0, ⋯ , κ − 1} for some fixed κ ∈{∞, 2, 3, ⋯ }. Assume that the Markovian evolution of the particle system (PS) is driven by some translation invariant local dynamics with bounded range, encoded by a jump rate matrix ⊤. These are standard settings, satisfied by the TASEP, the voter models, the contact processes. The aim of this paper is to provide some sufficient and/or necessary conditions on the matrix ⊤ so that this Markov process admits some simple invariant distribution, as a product measure (if 𝕃 is any of the spaces mentioned above), the law of a Markov process indexed by ℤ or [1, n] ∩ ℤ (if 𝕃 = ℤ or {1, …, n}), or a Gibbs measure if 𝕃 = ℤ/nℤ. Multiple applications follow: efficient ways to find invariant Markov laws for a given jump rate matrix or to prove that none exists. The voter models and the contact processes are shown not to possess any Markov laws as invariant distribution (for any memory m). (As usual, a random process X indexed by ℤ or ℕ is said to be a Markov chain with memory m ∈ {0, 1, 2, ⋯ } if ℙ(Xk ∈ A | Xk−i, i ≥ 1) = ℙ(Xk ∈ A | Xk−i, 1 ≤ i ≤ m), for any k.) We also prove that some models close to these models do. We exhibit PS admitting hidden Markov chains as invariant distribution and design many PS on ℤ2, with jump rates indexed by 2 × 2 squares, admitting product invariant measures.
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45

Svindland, Gregor. "Continuity properties of law-invariant (quasi-)convex risk functions on L ∞." Mathematics and Financial Economics 3, no. 1 (March 27, 2010): 39–43. http://dx.doi.org/10.1007/s11579-010-0026-x.

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46

He, Xue Dong, and Xianhua Peng. "Surplus-Invariant, Law-Invariant, and Conic Acceptance Sets Must Be the Sets Induced by Value at Risk." Operations Research 66, no. 5 (October 2018): 1268–75. http://dx.doi.org/10.1287/opre.2018.1743.

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47

Wedig, Walter V. "Invariant measures and Lyapunov exponents for generalized parameter fluctuations." Structural Safety 8, no. 1-4 (July 1990): 13–25. http://dx.doi.org/10.1016/0167-4730(90)90028-n.

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48

Filipović, Damir, and Gregor Svindland. "Optimal capital and risk allocations for law- and cash-invariant convex functions." Finance and Stochastics 12, no. 3 (May 29, 2008): 423–39. http://dx.doi.org/10.1007/s00780-008-0069-5.

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49

Carlier, G., and R. A. Dana. "Two-persons efficient risk-sharing and equilibria for concave law-invariant utilities." Economic Theory 36, no. 2 (July 17, 2007): 189–223. http://dx.doi.org/10.1007/s00199-007-0266-z.

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50

Kryzhevich, Sergey, Viktor Avrutin, Nikita Begun, Dmitrii Rachinskii, and Khosro Tajbakhsh. "Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps." Axioms 10, no. 2 (May 2, 2021): 80. http://dx.doi.org/10.3390/axioms10020080.

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We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and study its properties. Further, we study how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.
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