Pub Date : 2025-04-20DOI: 10.1016/j.spa.2025.104664
Mohamed Ben Alaya , Ahmed Kebaier , Gyula Pap , Ngoc Khue Tran
In this paper, we consider a one-dimensional jump-type Cox–Ingersoll–Ross process driven by a Brownian motion and a subordinator, whose growth rate is an unknown parameter. Considering the process observed continuously or discretely at high frequency, we derive the local asymptotic properties for the growth rate in both ergodic and non-ergodic cases. Local asymptotic normality (LAN) is proved in the subcritical case, local asymptotic quadraticity (LAQ) is derived in the critical case, and local asymptotic mixed normality (LAMN) is shown in the supercritical case. To obtain these results, techniques of Malliavin calculus and a subtle analysis on the jump structure of the subordinator involving the amplitude of jumps and number of jumps are essentially used.
{"title":"Local asymptotic properties for the growth rate of a jump-type CIR process","authors":"Mohamed Ben Alaya , Ahmed Kebaier , Gyula Pap , Ngoc Khue Tran","doi":"10.1016/j.spa.2025.104664","DOIUrl":"10.1016/j.spa.2025.104664","url":null,"abstract":"<div><div>In this paper, we consider a one-dimensional jump-type Cox–Ingersoll–Ross process driven by a Brownian motion and a subordinator, whose growth rate is an unknown parameter. Considering the process observed continuously or discretely at high frequency, we derive the local asymptotic properties for the growth rate in both ergodic and non-ergodic cases. Local asymptotic normality (LAN) is proved in the subcritical case, local asymptotic quadraticity (LAQ) is derived in the critical case, and local asymptotic mixed normality (LAMN) is shown in the supercritical case. To obtain these results, techniques of Malliavin calculus and a subtle analysis on the jump structure of the subordinator involving the amplitude of jumps and number of jumps are essentially used.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104664"},"PeriodicalIF":1.1,"publicationDate":"2025-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143867753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-19DOI: 10.1016/j.spa.2025.104652
Lucas R. de Lima , Daniel Valesin
This study delves into first-passage percolation on random geometric graphs in the supercritical regime, where the graphs exhibit a unique infinite connected component. We investigate properties such as geodesic paths, moderate deviations, and fluctuations, aiming to establish a quantitative shape theorem. Furthermore, we examine fluctuations in geodesic paths and characterize the properties of spanning trees and their semi-infinite paths.
{"title":"Speed of convergence and moderate deviations of FPP on random geometric graphs","authors":"Lucas R. de Lima , Daniel Valesin","doi":"10.1016/j.spa.2025.104652","DOIUrl":"10.1016/j.spa.2025.104652","url":null,"abstract":"<div><div>This study delves into first-passage percolation on random geometric graphs in the supercritical regime, where the graphs exhibit a unique infinite connected component. We investigate properties such as geodesic paths, moderate deviations, and fluctuations, aiming to establish a quantitative shape theorem. Furthermore, we examine fluctuations in geodesic paths and characterize the properties of spanning trees and their semi-infinite paths.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104652"},"PeriodicalIF":1.1,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143867754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-18DOI: 10.1016/j.spa.2025.104663
Yue Niu , Baoyou Qu , Falei Wang
The present paper focuses on the investigation of multi-dimensional mean reflected backward stochastic differential equations (BSDEs) in a possibly non-convex reflection domain, whose generator also depends on the marginal probability distributions of the solution . Our main idea is to decompose the mean reflected BSDE into a BSDE and a deterministic Skorokhod problem. Then, utilizing -estimates for BSDEs and Skorokhod problems, we explore the solvability of -solutions () through fixed-point argument and an approximation approach under both inward normal and oblique reflection scenarios.
{"title":"Lp-solutions of multi-dimensional BSDEs with mean reflection","authors":"Yue Niu , Baoyou Qu , Falei Wang","doi":"10.1016/j.spa.2025.104663","DOIUrl":"10.1016/j.spa.2025.104663","url":null,"abstract":"<div><div>The present paper focuses on the investigation of multi-dimensional mean reflected backward stochastic differential equations (BSDEs) in a possibly non-convex reflection domain, whose generator also depends on the marginal probability distributions of the solution <span><math><mrow><mo>(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>)</mo></mrow></math></span>. Our main idea is to decompose the mean reflected BSDE into a BSDE and a deterministic Skorokhod problem. Then, utilizing <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-estimates for BSDEs and Skorokhod problems, we explore the solvability of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-solutions (<span><math><mrow><mi>p</mi><mo>></mo><mn>1</mn></mrow></math></span>) through fixed-point argument and an approximation approach under both inward normal and oblique reflection scenarios.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104663"},"PeriodicalIF":1.1,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143851716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-17DOI: 10.1016/j.spa.2025.104662
Hongjiang Qian , Yanzhao Cao , George Yin
This paper develops large deviation estimates for nonlinear filtering with discontinuity in the drift of the signal and small noise intensities in both the signal and the observations. A variational approach related to Mortensen’s optimization problem is utilized in our analysis. The discontinuity of the drift in the signal naturally arises in many applications, including modeling communication channels with a “hard limiter”. Our results extend the work of Reddy et al. (2022), in which smooth functions were used. To address the discontinuous in the drift of the signal, relaxed controls are used to study the asymptotic fraction of time the controlled signals spend in each half-space divided by the discontinuity hyperplane. Large deviation estimates are established by the weak convergence method using the stochastic control representation for positive functionals of Brownian motions and Laplace asymptotics of the Kallianpur–Striebel formula.
{"title":"Large deviation estimates for nonlinear filtering with discontinuity and small noise","authors":"Hongjiang Qian , Yanzhao Cao , George Yin","doi":"10.1016/j.spa.2025.104662","DOIUrl":"10.1016/j.spa.2025.104662","url":null,"abstract":"<div><div>This paper develops large deviation estimates for nonlinear filtering with discontinuity in the drift of the signal and small noise intensities in both the signal and the observations. A variational approach related to Mortensen’s optimization problem is utilized in our analysis. The discontinuity of the drift in the signal naturally arises in many applications, including modeling communication channels with a “hard limiter”. Our results extend the work of Reddy et al. (2022), in which smooth functions were used. To address the discontinuous in the drift of the signal, relaxed controls are used to study the asymptotic fraction of time the controlled signals spend in each half-space divided by the discontinuity hyperplane. Large deviation estimates are established by the weak convergence method using the stochastic control representation for positive functionals of Brownian motions and Laplace asymptotics of the Kallianpur–Striebel formula.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104662"},"PeriodicalIF":1.1,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143843568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Over the past decade, the importance of the 1D signature which can be seen as a functional defined over a path, has been pivotal in both path-wise stochastic calculus and the analysis of time series data. By considering an image as a two-parameter function that takes values in a -dimensional space, we introduce an extension of the path signature to images. We address numerous challenges associated with this extension and demonstrate that the 2D signature satisfies a version of Chen’s relation in addition to a shuffle-type product. Furthermore, we show that specific variations of the 2D signature can be recursively defined, thereby satisfying an integral-type equation. We analyze the properties of the proposed signature, such as continuity, invariance to stretching, translation and rotation of the underlying image. Additionally, we establish that the proposed 2D signature over an image satisfies a universal approximation property.
{"title":"On the signature of an image","authors":"Joscha Diehl , Kurusch Ebrahimi-Fard , Fabian N. Harang , Samy Tindel","doi":"10.1016/j.spa.2025.104661","DOIUrl":"10.1016/j.spa.2025.104661","url":null,"abstract":"<div><div>Over the past decade, the importance of the 1D signature which can be seen as a functional defined over a path, has been pivotal in both path-wise stochastic calculus and the analysis of time series data. By considering an image as a two-parameter function that takes values in a <span><math><mi>d</mi></math></span>-dimensional space, we introduce an extension of the path signature to images. We address numerous challenges associated with this extension and demonstrate that the 2D signature satisfies a version of Chen’s relation in addition to a shuffle-type product. Furthermore, we show that specific variations of the 2D signature can be recursively defined, thereby satisfying an integral-type equation. We analyze the properties of the proposed signature, such as continuity, invariance to stretching, translation and rotation of the underlying image. Additionally, we establish that the proposed 2D signature over an image satisfies a universal approximation property.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104661"},"PeriodicalIF":1.1,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143868643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-15DOI: 10.1016/j.spa.2025.104654
Krzysztof Bogdan , Markus Kunze
We construct a Hunt process that can be described as an isotropic -stable Lévy process reflected from the complement of a bounded open Lipschitz set. In fact, we introduce a new analytic method for concatenating Markov processes. It is based on nonlocal Schrödinger perturbations of sub-Markovian transition kernels and the construction of two supermedian functions with different growth rates at infinity. We apply this framework to describe the return distribution and the stationary distribution of the process. To handle the strong Markov property at the reflection time, we introduce a novel ladder process, whose transition semigroup encodes not only the position of the process, but also the number of reflections.
{"title":"Stable processes with reflections","authors":"Krzysztof Bogdan , Markus Kunze","doi":"10.1016/j.spa.2025.104654","DOIUrl":"10.1016/j.spa.2025.104654","url":null,"abstract":"<div><div>We construct a Hunt process that can be described as an isotropic <span><math><mi>α</mi></math></span>-stable Lévy process reflected from the complement of a bounded open Lipschitz set. In fact, we introduce a new analytic method for concatenating Markov processes. It is based on nonlocal Schrödinger perturbations of sub-Markovian transition kernels and the construction of two supermedian functions with different growth rates at infinity. We apply this framework to describe the return distribution and the stationary distribution of the process. To handle the strong Markov property at the reflection time, we introduce a novel ladder process, whose transition semigroup encodes not only the position of the process, but also the number of reflections.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104654"},"PeriodicalIF":1.1,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143843544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-13DOI: 10.1016/j.spa.2025.104653
Yuanping Cui , Xiaoyue Li , Xuerong Mao
This manuscript is dedicated to the numerical approximation of super-linear slow-fast stochastic differential equations (SFSDEs). Borrowing the heterogeneous multiscale idea, we propose an explicit multiscale Euler–Maruyama scheme suitable for SFSDEs with locally Lipschitz coefficients using an appropriate truncation technique. By the averaging principle, we establish the strong convergence of the numerical solutions to the exact solutions in the th moment. Additionally, under lenient conditions on the coefficients, we also furnish a strong error estimate. In conclusion, we give two illustrative examples and accompanying numerical simulations to affirm the theoretical outcomes.
{"title":"Explicit multiscale numerical method for super-linear slow-fast stochastic differential equations","authors":"Yuanping Cui , Xiaoyue Li , Xuerong Mao","doi":"10.1016/j.spa.2025.104653","DOIUrl":"10.1016/j.spa.2025.104653","url":null,"abstract":"<div><div>This manuscript is dedicated to the numerical approximation of super-linear slow-fast stochastic differential equations (SFSDEs). Borrowing the heterogeneous multiscale idea, we propose an explicit multiscale Euler–Maruyama scheme suitable for SFSDEs with locally Lipschitz coefficients using an appropriate truncation technique. By the averaging principle, we establish the strong convergence of the numerical solutions to the exact solutions in the <span><math><mi>p</mi></math></span>th moment. Additionally, under lenient conditions on the coefficients, we also furnish a strong error estimate. In conclusion, we give two illustrative examples and accompanying numerical simulations to affirm the theoretical outcomes.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104653"},"PeriodicalIF":1.1,"publicationDate":"2025-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143843656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-12DOI: 10.1016/j.spa.2025.104651
Stefan Geiss , Nguyen Tran Thuan
We discuss in a stochastic framework the interplay between Riemann–Liouville type operators applied to stochastic processes, bounded mean oscillation, real interpolation, and approximation. In particular, we investigate the singularity of gradient processes on the Wiener space arising from parabolic PDEs via the Feynman–Kac theory. The singularity is measured in terms of bmo-conditions on the fractional integrated gradient. As an application we treat an approximation problem for stochastic integrals on the Wiener space. In particular, we provide a discrete time hedging strategy for the binary option with a uniform local control of the hedging error under a shortfall constraint.
{"title":"On Riemann–Liouville type operators, bounded mean oscillation, gradient estimates and approximation on the Wiener space","authors":"Stefan Geiss , Nguyen Tran Thuan","doi":"10.1016/j.spa.2025.104651","DOIUrl":"10.1016/j.spa.2025.104651","url":null,"abstract":"<div><div>We discuss in a stochastic framework the interplay between Riemann–Liouville type operators applied to stochastic processes, bounded mean oscillation, real interpolation, and approximation. In particular, we investigate the singularity of gradient processes on the Wiener space arising from parabolic PDEs via the Feynman–Kac theory. The singularity is measured in terms of bmo-conditions on the fractional integrated gradient. As an application we treat an approximation problem for stochastic integrals on the Wiener space. In particular, we provide a discrete time hedging strategy for the binary option with a uniform local control of the hedging error under a shortfall constraint.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104651"},"PeriodicalIF":1.1,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143843569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We address parameter estimation in second-order stochastic differential equations (SDEs), which are prevalent in physics, biology, and ecology. The second-order SDE is converted to a first-order system by introducing an auxiliary velocity variable, which raises two main challenges. First, the system is hypoelliptic since the noise affects only the velocity, making the Euler–Maruyama estimator ill-conditioned. We propose an estimator based on the Strang splitting scheme to overcome this. Second, since the velocity is rarely observed, we adapt the estimator to partial observations. We present four estimators for complete and partial observations, using the full pseudo-likelihood or only the velocity-based partial pseudo-likelihood. These estimators are intuitive, easy to implement, and computationally fast, and we prove their consistency and asymptotic normality. Our analysis demonstrates that using the full pseudo-likelihood with complete observations reduces the asymptotic variance of the diffusion estimator. With partial observations, the asymptotic variance increases as a result of information loss but remains unaffected by the likelihood choice. However, a numerical study on the Kramers oscillator reveals that using the partial pseudo-likelihood for partial observations yields less biased estimators. We apply our approach to paleoclimate data from the Greenland ice core by fitting the Kramers oscillator model, capturing transitions between metastable states reflecting observed climatic conditions during glacial eras.
{"title":"Strang splitting for parametric inference in second-order stochastic differential equations","authors":"Predrag Pilipovic , Adeline Samson , Susanne Ditlevsen","doi":"10.1016/j.spa.2025.104650","DOIUrl":"10.1016/j.spa.2025.104650","url":null,"abstract":"<div><div>We address parameter estimation in second-order stochastic differential equations (SDEs), which are prevalent in physics, biology, and ecology. The second-order SDE is converted to a first-order system by introducing an auxiliary velocity variable, which raises two main challenges. First, the system is hypoelliptic since the noise affects only the velocity, making the Euler–Maruyama estimator ill-conditioned. We propose an estimator based on the Strang splitting scheme to overcome this. Second, since the velocity is rarely observed, we adapt the estimator to partial observations. We present four estimators for complete and partial observations, using the full pseudo-likelihood or only the velocity-based partial pseudo-likelihood. These estimators are intuitive, easy to implement, and computationally fast, and we prove their consistency and asymptotic normality. Our analysis demonstrates that using the full pseudo-likelihood with complete observations reduces the asymptotic variance of the diffusion estimator. With partial observations, the asymptotic variance increases as a result of information loss but remains unaffected by the likelihood choice. However, a numerical study on the Kramers oscillator reveals that using the partial pseudo-likelihood for partial observations yields less biased estimators. We apply our approach to paleoclimate data from the Greenland ice core by fitting the Kramers oscillator model, capturing transitions between metastable states reflecting observed climatic conditions during glacial eras.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104650"},"PeriodicalIF":1.1,"publicationDate":"2025-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143829599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-07DOI: 10.1016/j.spa.2025.104648
Pascal Oswald
We study the ancestral lineages of individuals of a stationary discrete-time branching annihilating random walk (BARW) on the -dimensional lattice . Each individual produces a Poissonian number of offspring with mean which then jump independently to a uniformly chosen site with a fixed distance of their parent. Should two or more particles jump to the same site, all particles at that site get annihilated. By interpreting the ancestral lineage of such an individual as a random walk in a dynamical random environment, we obtain a law of large numbers and a functional central limit theorem for the ancestral lineage whenever the model parameters satisfy and is large enough.
{"title":"Ancestral lineages for a branching annihilating random walk","authors":"Pascal Oswald","doi":"10.1016/j.spa.2025.104648","DOIUrl":"10.1016/j.spa.2025.104648","url":null,"abstract":"<div><div>We study the ancestral lineages of individuals of a stationary discrete-time branching annihilating random walk (BARW) on the <span><math><mi>d</mi></math></span>-dimensional lattice <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. Each individual produces a Poissonian number of offspring with mean <span><math><mi>μ</mi></math></span> which then jump independently to a uniformly chosen site with a fixed distance <span><math><mi>R</mi></math></span> of their parent. Should two or more particles jump to the same site, all particles at that site get annihilated. By interpreting the ancestral lineage of such an individual as a random walk in a dynamical random environment, we obtain a law of large numbers and a functional central limit theorem for the ancestral lineage whenever the model parameters satisfy <span><math><mrow><mi>μ</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>R</mi><mo>=</mo><mi>R</mi><mrow><mo>(</mo><mi>μ</mi><mo>)</mo></mrow></mrow></math></span> is large enough.</div></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"187 ","pages":"Article 104648"},"PeriodicalIF":1.1,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143835136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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