Abstract We establish a pathwise large deviation principle for affine stochastic volatility models introduced by Keller-Ressel (2011), and present an application to variance reduction for Monte Carlo computation of prices of path-dependent options in these models, extending the method developed by Genin and Tankov (2020) for exponential Lévy models. To this end, we apply an exponentially affine change of measure and use Varadhan’s lemma, in the fashion of Guasoni and Robertson (2008) and Robertson (2010), to approximate the problem of finding the measure that minimizes the variance of the Monte Carlo estimator. We test the method on the Heston model with and without jumps to demonstrate its numerical efficiency.
{"title":"Long-Time Trajectorial Large Deviations and Importance Sampling for Affine Stochastic Volatility Models","authors":"Z. Grbac, David Krief, P. Tankov","doi":"10.1017/apr.2020.58","DOIUrl":"https://doi.org/10.1017/apr.2020.58","url":null,"abstract":"Abstract We establish a pathwise large deviation principle for affine stochastic volatility models introduced by Keller-Ressel (2011), and present an application to variance reduction for Monte Carlo computation of prices of path-dependent options in these models, extending the method developed by Genin and Tankov (2020) for exponential Lévy models. To this end, we apply an exponentially affine change of measure and use Varadhan’s lemma, in the fashion of Guasoni and Robertson (2008) and Robertson (2010), to approximate the problem of finding the measure that minimizes the variance of the Monte Carlo estimator. We test the method on the Heston model with and without jumps to demonstrate its numerical efficiency.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"53 1","pages":"220 - 250"},"PeriodicalIF":1.2,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/apr.2020.58","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45860766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study the distributions of component and system lifetimes under the time-homogeneous load-sharing model, where the multivariate conditional hazard rates of working components depend only on the set of failed components, and not on their failure moments or the time elapsed from the start of system operation. Then we analyze its time-heterogeneous extension, in which the distributions of consecutive failure times, single component lifetimes, and system lifetimes coincide with mixtures of distributions of generalized order statistics. Finally we focus on some specific forms of the time-nonhomogeneous load-sharing model.
{"title":"Load-Sharing Reliability Models with Different Component Sensitivities to Other Components’ Working States","authors":"T. Rychlik, F. Spizzichino","doi":"10.1017/apr.2020.49","DOIUrl":"https://doi.org/10.1017/apr.2020.49","url":null,"abstract":"Abstract We study the distributions of component and system lifetimes under the time-homogeneous load-sharing model, where the multivariate conditional hazard rates of working components depend only on the set of failed components, and not on their failure moments or the time elapsed from the start of system operation. Then we analyze its time-heterogeneous extension, in which the distributions of consecutive failure times, single component lifetimes, and system lifetimes coincide with mixtures of distributions of generalized order statistics. Finally we focus on some specific forms of the time-nonhomogeneous load-sharing model.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"53 1","pages":"107 - 132"},"PeriodicalIF":1.2,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/apr.2020.49","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46521333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider Poisson hail models and characterize up to boundaries the collection of critical moments which guarantee stability. In particular, we treat the case of infinite speed of propagation.
我们考虑泊松冰雹模型，并将保证稳定性的临界矩集合刻画到边界。特别是，我们处理无限传播速度的情况。
{"title":"Stable systems with power law conditions for Poisson hail","authors":"T. Mountford, Zhe Wang","doi":"10.1017/apr.2023.23","DOIUrl":"https://doi.org/10.1017/apr.2023.23","url":null,"abstract":"\u0000 We consider Poisson hail models and characterize up to boundaries the collection of critical moments which guarantee stability. In particular, we treat the case of infinite speed of propagation.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48528879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Crisan, P. Del Moral, A. Jasra, Hamza M. Ruzayqat
Abstract In this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman–Bucy filter estimates based upon several nonlinear Kalman–Bucy diffusions. Using new conditional bias results for the mean of the aforementioned methods, we analyze the empirical log-scale normalization constants in terms of their $mathbb{L}_n$ -errors and $mathbb{L}_n$ -conditional bias. Depending on the type of nonlinear Kalman–Bucy diffusion, we show that these are bounded above by terms such as $mathsf{C}(n)left[t^{1/2}/N^{1/2} + t/Nright]$ or $mathsf{C}(n)/N^{1/2}$ ( $mathbb{L}_n$ -errors) and $mathsf{C}(n)left[t+t^{1/2}right]/N$ or $mathsf{C}(n)/N$ ( $mathbb{L}_n$ -conditional bias), where t is the time horizon, N is the ensemble size, and $mathsf{C}(n)$ is a constant that depends only on n, not on N or t. Finally, we use these results for online static parameter estimation for the above filtering models and implement the methodology for both linear and nonlinear models.
{"title":"Log-normalization constant estimation using the ensemble Kalman–Bucy filter with application to high-dimensional models","authors":"D. Crisan, P. Del Moral, A. Jasra, Hamza M. Ruzayqat","doi":"10.1017/apr.2021.62","DOIUrl":"https://doi.org/10.1017/apr.2021.62","url":null,"abstract":"Abstract In this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman–Bucy filter estimates based upon several nonlinear Kalman–Bucy diffusions. Using new conditional bias results for the mean of the aforementioned methods, we analyze the empirical log-scale normalization constants in terms of their \u0000$mathbb{L}_n$\u0000 -errors and \u0000$mathbb{L}_n$\u0000 -conditional bias. Depending on the type of nonlinear Kalman–Bucy diffusion, we show that these are bounded above by terms such as \u0000$mathsf{C}(n)left[t^{1/2}/N^{1/2} + t/Nright]$\u0000 or \u0000$mathsf{C}(n)/N^{1/2}$\u0000 ( \u0000$mathbb{L}_n$\u0000 -errors) and \u0000$mathsf{C}(n)left[t+t^{1/2}right]/N$\u0000 or \u0000$mathsf{C}(n)/N$\u0000 ( \u0000$mathbb{L}_n$\u0000 -conditional bias), where t is the time horizon, N is the ensemble size, and \u0000$mathsf{C}(n)$\u0000 is a constant that depends only on n, not on N or t. Finally, we use these results for online static parameter estimation for the above filtering models and implement the methodology for both linear and nonlinear models.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"54 1","pages":"1139 - 1163"},"PeriodicalIF":1.2,"publicationDate":"2021-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49061485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abraham Gutierrez, Sebastian Müller, Stjepan Šebek
Abstract The basic idea of voting protocols is that nodes query a sample of other nodes and adjust their own opinion throughout several rounds based on the proportion of the sampled opinions. In the classic model, it is assumed that all nodes have the same weight. We study voting protocols for heterogeneous weights with respect to fairness. A voting protocol is fair if the influence on the eventual outcome of a given participant is linear in its weight. Previous work used sampling with replacement to construct a fair voting scheme. However, it was shown that using greedy sampling, i.e., sampling with replacement until a given number of distinct elements is chosen, turns out to be more robust and performant. In this paper, we study fairness of voting protocols with greedy sampling and propose a voting scheme that is asymptotically fair for a broad class of weight distributions. We complement our theoretical findings with numerical results and present several open questions and conjectures.
{"title":"On asymptotic fairness in voting with greedy sampling","authors":"Abraham Gutierrez, Sebastian Müller, Stjepan Šebek","doi":"10.1017/apr.2022.63","DOIUrl":"https://doi.org/10.1017/apr.2022.63","url":null,"abstract":"Abstract The basic idea of voting protocols is that nodes query a sample of other nodes and adjust their own opinion throughout several rounds based on the proportion of the sampled opinions. In the classic model, it is assumed that all nodes have the same weight. We study voting protocols for heterogeneous weights with respect to fairness. A voting protocol is fair if the influence on the eventual outcome of a given participant is linear in its weight. Previous work used sampling with replacement to construct a fair voting scheme. However, it was shown that using greedy sampling, i.e., sampling with replacement until a given number of distinct elements is chosen, turns out to be more robust and performant. In this paper, we study fairness of voting protocols with greedy sampling and propose a voting scheme that is asymptotically fair for a broad class of weight distributions. We complement our theoretical findings with numerical results and present several open questions and conjectures.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"55 1","pages":"999 - 1032"},"PeriodicalIF":1.2,"publicationDate":"2021-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44487107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we construct operator fractional Lévy motion (ofLm), a broad class of infinitely divisible stochastic processes that are covariance operator self-similar and have wide-sense stationary increments. The ofLm class generalizes the univariate fractional Lévy motion as well as the multivariate operator fractional Brownian motion (ofBm). OfLm can be divided into two types, namely, moving average (maofLm) and real harmonizable (rhofLm), both of which share the covariance structure of ofBm under assumptions. We show that maofLm and rhofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small- and large-scale limiting behavior. We also characterize time-reversibility for ofLm through parametric conditions related to its Lévy measure. In particular, we show that, under non-Gaussianity, the parametric conditions for time-reversibility are generally more restrictive than those for the Gaussian case (ofBm).
{"title":"On operator fractional Lévy motion: integral representations and time-reversibility","authors":"B. C. Boniece, G. Didier","doi":"10.1017/apr.2021.41","DOIUrl":"https://doi.org/10.1017/apr.2021.41","url":null,"abstract":"Abstract In this paper, we construct operator fractional Lévy motion (ofLm), a broad class of infinitely divisible stochastic processes that are covariance operator self-similar and have wide-sense stationary increments. The ofLm class generalizes the univariate fractional Lévy motion as well as the multivariate operator fractional Brownian motion (ofBm). OfLm can be divided into two types, namely, moving average (maofLm) and real harmonizable (rhofLm), both of which share the covariance structure of ofBm under assumptions. We show that maofLm and rhofLm admit stochastic integral representations in the time and Fourier domains, and establish their distinct small- and large-scale limiting behavior. We also characterize time-reversibility for ofLm through parametric conditions related to its Lévy measure. In particular, we show that, under non-Gaussianity, the parametric conditions for time-reversibility are generally more restrictive than those for the Gaussian case (ofBm).","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"54 1","pages":"493 - 535"},"PeriodicalIF":1.2,"publicationDate":"2021-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44804814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
G. di Nunno, Y. Mishura, Anton Yurchenko-Tytarenko
Abstract We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of order $lambda in (0,1)$ . The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some mild assumptions on the noise, we prove that the solution has moments of all orders. In addition, we provide its connection to the solution of some Skorokhod reflection problem. As an illustration of our results and motivation for applications, we also suggest two stochastic volatility models which we regard as generalizations of the CIR and CEV processes. We complete the study by providing a numerical scheme for the solution.
{"title":"Sandwiched SDEs with unbounded drift driven by Hölder noises","authors":"G. di Nunno, Y. Mishura, Anton Yurchenko-Tytarenko","doi":"10.1017/apr.2022.56","DOIUrl":"https://doi.org/10.1017/apr.2022.56","url":null,"abstract":"Abstract We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of order \u0000$lambda in (0,1)$\u0000 . The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some mild assumptions on the noise, we prove that the solution has moments of all orders. In addition, we provide its connection to the solution of some Skorokhod reflection problem. As an illustration of our results and motivation for applications, we also suggest two stochastic volatility models which we regard as generalizations of the CIR and CEV processes. We complete the study by providing a numerical scheme for the solution.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"55 1","pages":"927 - 964"},"PeriodicalIF":1.2,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42175319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A system of interacting multi-class finite-state jump processes is analyzed. The model under consideration consists of a block-structured network with dynamically changing multi-color nodes. The interactions are local and described through local empirical measures. Two levels of heterogeneity are considered: between and within the blocks where the nodes are labeled into two types. The central nodes are those connected only to nodes from the same block, whereas the peripheral nodes are connected to both nodes from the same block and nodes from other blocks. Limits of such systems as the number of nodes tends to infinity are investigated. In particular, under specific regularity conditions, propagation of chaos and the law of large numbers are established in a multi-population setting. Moreover, it is shown that, as the number of nodes goes to infinity, the behavior of the system can be represented by the solution of a McKean–Vlasov system. Then, we prove large deviations principles for the vectors of empirical measures and the empirical processes, which extends the classical results of Dawson and Gärtner (Stochastics20, 1987) and Léonard (Ann. Inst. H. Poincaré Prob. Statist.31, 1995).
{"title":"Propagation of chaos and large deviations in mean-field models with jumps on block-structured networks","authors":"D. Dawson, Ahmed Sid-Ali, Yiqiang Q. Zhao","doi":"10.1017/apr.2023.7","DOIUrl":"https://doi.org/10.1017/apr.2023.7","url":null,"abstract":"\u0000 A system of interacting multi-class finite-state jump processes is analyzed. The model under consideration consists of a block-structured network with dynamically changing multi-color nodes. The interactions are local and described through local empirical measures. Two levels of heterogeneity are considered: between and within the blocks where the nodes are labeled into two types. The central nodes are those connected only to nodes from the same block, whereas the peripheral nodes are connected to both nodes from the same block and nodes from other blocks. Limits of such systems as the number of nodes tends to infinity are investigated. In particular, under specific regularity conditions, propagation of chaos and the law of large numbers are established in a multi-population setting. Moreover, it is shown that, as the number of nodes goes to infinity, the behavior of the system can be represented by the solution of a McKean–Vlasov system. Then, we prove large deviations principles for the vectors of empirical measures and the empirical processes, which extends the classical results of Dawson and Gärtner (Stochastics20, 1987) and Léonard (Ann. Inst. H. Poincaré Prob. Statist.31, 1995).","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2020-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43027331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The study of multivariate extremes is dominated by multivariate regular variation, although it is well known that this approach does not provide adequate distinction between random vectors whose components are not always simultaneously large. Various alternative dependence measures and representations have been proposed, with the most well-known being hidden regular variation and the conditional extreme value model. These varying depictions of extremal dependence arise through consideration of different parts of the multivariate domain, and particularly through exploring what happens when extremes of one variable may grow at different rates from other variables. Thus far, these alternative representations have come from distinct sources, and links between them are limited. In this work we elucidate many of the relevant connections through a geometrical approach. In particular, the shape of the limit set of scaled sample clouds in light-tailed margins is shown to provide a description of several different extremal dependence representations.
{"title":"Linking representations for multivariate extremes via a limit set","authors":"N. Nolde, J. Wadsworth","doi":"10.1017/apr.2021.51","DOIUrl":"https://doi.org/10.1017/apr.2021.51","url":null,"abstract":"Abstract The study of multivariate extremes is dominated by multivariate regular variation, although it is well known that this approach does not provide adequate distinction between random vectors whose components are not always simultaneously large. Various alternative dependence measures and representations have been proposed, with the most well-known being hidden regular variation and the conditional extreme value model. These varying depictions of extremal dependence arise through consideration of different parts of the multivariate domain, and particularly through exploring what happens when extremes of one variable may grow at different rates from other variables. Thus far, these alternative representations have come from distinct sources, and links between them are limited. In this work we elucidate many of the relevant connections through a geometrical approach. In particular, the shape of the limit set of scaled sample clouds in light-tailed margins is shown to provide a description of several different extremal dependence representations.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"54 1","pages":"688 - 717"},"PeriodicalIF":1.2,"publicationDate":"2020-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45477623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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