Abstract In this paper, we study the analytical properties of the true solution to the generalised delay Ait-Sahalia-type interest rate model with Poisson-driven jumps. Since this model does not have a closed-form solution, we employ several new truncated Euler-Maruyama (EM) techniques to investigate the finite-time strong convergence theory of the numerical solution under the local Lipschitz condition plus the Khasminskii-type condition. We justify the strong convergence result for Monte Carlo calibration and valuation of some debt and derivative instruments.
{"title":"Delay Ait-Sahalia-type interest rate model with jumps and its strong approximation","authors":"E. Coffie","doi":"10.1515/strm-2022-0013","DOIUrl":"https://doi.org/10.1515/strm-2022-0013","url":null,"abstract":"Abstract In this paper, we study the analytical properties of the true solution to the generalised delay Ait-Sahalia-type interest rate model with Poisson-driven jumps. Since this model does not have a closed-form solution, we employ several new truncated Euler-Maruyama (EM) techniques to investigate the finite-time strong convergence theory of the numerical solution under the local Lipschitz condition plus the Khasminskii-type condition. We justify the strong convergence result for Monte Carlo calibration and valuation of some debt and derivative instruments.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43664596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We propose to derive deviation measures through the Minkowski gauge of a given set of acceptable positions. We show that, given a suitable acceptance set, any positive homogeneous deviation measure can be accommodated in our framework. In doing so, we provide a new interpretation for such measures, namely, that they quantify how much one must shrink or deleverage a financial position for it to become acceptable. In particular, the Minkowski Deviation of a set which is convex, translation insensitive, and radially bounded at non-constants, is a generalized deviation measure in the sense of [R. T. Rockafellar, S. Uryasev and M. Zabarankin, Generalized deviations in risk analysis, Finance Stoch. 10 2006, 1, 51–74]. Furthermore, we explore the converse relations from properties of a Minkowski Deviation to its sub-level sets, introducing the notion of acceptance sets for deviations. Hence, we fill a gap existing in the literature, namely the lack of a well-defined concept of acceptance sets for deviation measures. Dual characterizations in terms of polar sets and support functionals are provided.
{"title":"Minkowski deviation measures","authors":"Marlon Moresco, Marcelo Brutti Righi, E. Horta","doi":"10.1515/strm-2021-0033","DOIUrl":"https://doi.org/10.1515/strm-2021-0033","url":null,"abstract":"Abstract We propose to derive deviation measures through the Minkowski gauge of a given set of acceptable positions. We show that, given a suitable acceptance set, any positive homogeneous deviation measure can be accommodated in our framework. In doing so, we provide a new interpretation for such measures, namely, that they quantify how much one must shrink or deleverage a financial position for it to become acceptable. In particular, the Minkowski Deviation of a set which is convex, translation insensitive, and radially bounded at non-constants, is a generalized deviation measure in the sense of [R. T. Rockafellar, S. Uryasev and M. Zabarankin, Generalized deviations in risk analysis, Finance Stoch. 10 2006, 1, 51–74]. Furthermore, we explore the converse relations from properties of a Minkowski Deviation to its sub-level sets, introducing the notion of acceptance sets for deviations. Hence, we fill a gap existing in the literature, namely the lack of a well-defined concept of acceptance sets for deviation measures. Dual characterizations in terms of polar sets and support functionals are provided.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47013827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we explore the idea of grouping under the massive data framework, to propose a median-of-means non-parametric type estimator for the Proportional Hazard Transform (PHT), which has been widely used in finance and insurance. Under certain conditions on the growth rate of subgroups, the consistency and asymptotic normality of the proposed estimators are investigated. Furthermore, we construct a new method to test PHT based on the empirical likelihood method for the median in order to avoid any prior estimate of the variance structure for the proposed estimator, as it is difficult to estimate and often causes much inaccuracy. Numerical simulations and real-data analysis are designed to show the present estimator’s performance. The results confirm that the new put-forward estimator is quite robust with respect to outliers.
{"title":"A robust estimator of the proportional hazard transform for massive data","authors":"Tami Omar, Rassoul Abdelaziz, Ould Rouis Hamid","doi":"10.1515/strm-2020-0007","DOIUrl":"https://doi.org/10.1515/strm-2020-0007","url":null,"abstract":"Abstract In this paper, we explore the idea of grouping under the massive data framework, to propose a median-of-means non-parametric type estimator for the Proportional Hazard Transform (PHT), which has been widely used in finance and insurance. Under certain conditions on the growth rate of subgroups, the consistency and asymptotic normality of the proposed estimators are investigated. Furthermore, we construct a new method to test PHT based on the empirical likelihood method for the median in order to avoid any prior estimate of the variance structure for the proposed estimator, as it is difficult to estimate and often causes much inaccuracy. Numerical simulations and real-data analysis are designed to show the present estimator’s performance. The results confirm that the new put-forward estimator is quite robust with respect to outliers.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49155229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract To increase the power of the VaR tests, it has been recently proposed to extend the duration-based test class with the geometric-VaR and Gini-coefficient-based tests. These tests, though exhibiting outstanding power properties, have not gained recognition in the industry. A potential reason is the absence of ready-to-use statistical distributions. To remedy this, we inquire into the limiting properties of these tests and derive relevant asymptotic distributions. We also provide a generalized geometric-VaR test and give its distribution. Through the Monte Carlo study, we show the accuracy of our asymptotic procedures in finite samples, and we find these procedures to be relevant for the current Basel standards. Our theoretical results are illustrated by the empirical study that includes data from the current COVID-19 crisis.
{"title":"Asymptotic properties of duration-based VaR backtests","authors":"Marta Małecka","doi":"10.1515/strm-2021-0019","DOIUrl":"https://doi.org/10.1515/strm-2021-0019","url":null,"abstract":"Abstract To increase the power of the VaR tests, it has been recently proposed to extend the duration-based test class with the geometric-VaR and Gini-coefficient-based tests. These tests, though exhibiting outstanding power properties, have not gained recognition in the industry. A potential reason is the absence of ready-to-use statistical distributions. To remedy this, we inquire into the limiting properties of these tests and derive relevant asymptotic distributions. We also provide a generalized geometric-VaR test and give its distribution. Through the Monte Carlo study, we show the accuracy of our asymptotic procedures in finite samples, and we find these procedures to be relevant for the current Basel standards. Our theoretical results are illustrated by the empirical study that includes data from the current COVID-19 crisis.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43536156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Phase-type distributions are dense in the class of distributions on the positive real line, and their flexibility and closed-form formulas in terms of matrix calculus allow fitting models to data in various application areas. However, the parameters are in general non-identifiable, and hence the dimension of two similar models may be very different. This paper proposes a new method for selecting the dimension of phase-type distributions via penalisation of the likelihood function. The penalties are in terms of the Green matrix, from which it is possible to extract the contributions of each state to the overall mean. Since representations with higher dimensions are penalised, a parsimony effect is obtained. We perform a numerical study with randomly generated phase-type samples to illustrate the effectiveness of the proposed procedure, and also apply the technique to the absolute log-returns of EURO STOXX 50 and Bitcoin prices.
{"title":"Penalised likelihood methods for phase-type dimension selection","authors":"H. Albrecher, Martin Bladt, Alaric J. A. Müller","doi":"10.1515/strm-2021-0026","DOIUrl":"https://doi.org/10.1515/strm-2021-0026","url":null,"abstract":"Abstract Phase-type distributions are dense in the class of distributions on the positive real line, and their flexibility and closed-form formulas in terms of matrix calculus allow fitting models to data in various application areas. However, the parameters are in general non-identifiable, and hence the dimension of two similar models may be very different. This paper proposes a new method for selecting the dimension of phase-type distributions via penalisation of the likelihood function. The penalties are in terms of the Green matrix, from which it is possible to extract the contributions of each state to the overall mean. Since representations with higher dimensions are penalised, a parsimony effect is obtained. We perform a numerical study with randomly generated phase-type samples to illustrate the effectiveness of the proposed procedure, and also apply the technique to the absolute log-returns of EURO STOXX 50 and Bitcoin prices.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42230889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yichen Feng, J. Fouque, Ruimeng Hu, Tomoyuki Ichiba
Abstract We analyze the systemic risk for disjoint and overlapping groups of financial institutions by proposing new models with realistic game features. Specifically, we generalize the systemic risk measure proposed in [F. Biagini, J.-P. Fouque, M. Frittelli and T. Meyer-Brandis, On fairness of systemic risk measures, Finance Stoch. 24 (2020), 2, 513–564] by allowing individual banks to choose their preferred groups instead of being assigned to certain groups. We introduce the concept of Nash equilibrium for these new models, and analyze the optimal solution under Gaussian distribution of the risk factor. We also provide an explicit solution for the risk allocation of the individual banks and study the existence and uniqueness of Nash equilibrium both theoretically and numerically. The developed numerical algorithm can simulate scenarios of equilibrium, and we apply it to study the banking structure with real data and show the validity of the proposed model.
摘要我们通过提出具有现实博弈特征的新模型来分析不相交和重叠的金融机构群体的系统风险。具体而言，我们通过允许个别银行选择自己喜欢的群体，而不是被分配到某些群体，来推广[F.Biagini，J.-P.Fouque，M.Fritelli和T.Meyer Brandis，On fairity of systemic risk measures，Finance Stoch.24（2020），2513–564]中提出的系统性风险测度。我们为这些新模型引入了纳什均衡的概念，并分析了风险因子在高斯分布下的最优解。我们还为单个银行的风险分配提供了一个显式的解决方案，并从理论和数值上研究了纳什均衡的存在性和唯一性。所开发的数值算法可以模拟均衡情景，并将其应用于实际数据的银行结构研究，验证了所提出模型的有效性。
{"title":"Systemic risk models for disjoint and overlapping groups with equilibrium strategies","authors":"Yichen Feng, J. Fouque, Ruimeng Hu, Tomoyuki Ichiba","doi":"10.1515/strm-2022-0004","DOIUrl":"https://doi.org/10.1515/strm-2022-0004","url":null,"abstract":"Abstract We analyze the systemic risk for disjoint and overlapping groups of financial institutions by proposing new models with realistic game features. Specifically, we generalize the systemic risk measure proposed in [F. Biagini, J.-P. Fouque, M. Frittelli and T. Meyer-Brandis, On fairness of systemic risk measures, Finance Stoch. 24 (2020), 2, 513–564] by allowing individual banks to choose their preferred groups instead of being assigned to certain groups. We introduce the concept of Nash equilibrium for these new models, and analyze the optimal solution under Gaussian distribution of the risk factor. We also provide an explicit solution for the risk allocation of the individual banks and study the existence and uniqueness of Nash equilibrium both theoretically and numerically. The developed numerical algorithm can simulate scenarios of equilibrium, and we apply it to study the banking structure with real data and show the validity of the proposed model.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45250405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract One of the important issues is risk assessment and calculation in complex and multi-component systems. In this paper, the estimation of multi-component stress-strength reliability for the Weibull distribution under the progressive Type-II censored samples is studied. We assume that both stress and strength are two independent Weibull distributions with different parameters. First, assuming the same shape parameter, the maximum likelihood estimation (MLE), different approximations of Bayes estimators (Lindley’s approximation and Markov chain Monte Carlo method) and different confidence intervals (asymptotic and highest posterior density) are obtained. In the case when the shape parameter is known, the MLE, uniformly minimum variance unbiased estimator (UMVUE), exact Bayes estimator and different confidence intervals (asymptotic and highest posterior density) are considered. Finally, in the general case, the statistical inferences on multi-component stress-strength reliability are derived. To compare the performances of different methods, Monte Carlo simulations are performed. Moreover, one data set for illustrative purposes is analyzed.
{"title":"Multi-component stress-strength model for Weibull distribution in progressively censored samples","authors":"A. Kohansal, S. Shoaee, S. Nadarajah","doi":"10.1515/strm-2020-0030","DOIUrl":"https://doi.org/10.1515/strm-2020-0030","url":null,"abstract":"Abstract One of the important issues is risk assessment and calculation in complex and multi-component systems. In this paper, the estimation of multi-component stress-strength reliability for the Weibull distribution under the progressive Type-II censored samples is studied. We assume that both stress and strength are two independent Weibull distributions with different parameters. First, assuming the same shape parameter, the maximum likelihood estimation (MLE), different approximations of Bayes estimators (Lindley’s approximation and Markov chain Monte Carlo method) and different confidence intervals (asymptotic and highest posterior density) are obtained. In the case when the shape parameter is known, the MLE, uniformly minimum variance unbiased estimator (UMVUE), exact Bayes estimator and different confidence intervals (asymptotic and highest posterior density) are considered. Finally, in the general case, the statistical inferences on multi-component stress-strength reliability are derived. To compare the performances of different methods, Monte Carlo simulations are performed. Moreover, one data set for illustrative purposes is analyzed.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46185321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider a Lévy driven stochastic convolution, also called continuous time Lévy driven moving average model X(t)=∫0ta(t-s)dZ(s)X(t)=int_{0}^{t}a(t-s),dZ(s), where 𝑍 is a Lévy martingale and the kernel a(.)a(,{.},) a deterministic function square integrable on R+mathbb{R}^{+}. Given 𝑁 i.i.d. continuous time observations (Xi(t))t∈[0,T](X_{i}(t))_{tin[0,T]}, i=1,…,Ni=1,dots,N, distributed like (X(t))t∈[0,T](X(t))_{tin[0,T]}, we propose two types of nonparametric projection estimators of a2a^{2} under different sets of assumptions. We bound the L2mathbb{L}^{2}-risk of the estimators and propose a data driven procedure to select the dimension of the projection space, illustrated by a short simulation study.
{"title":"Kernel estimation for Lévy driven stochastic convolutions","authors":"F. Comte, V. Genon-Catalot","doi":"10.1515/strm-2021-0007","DOIUrl":"https://doi.org/10.1515/strm-2021-0007","url":null,"abstract":"Abstract We consider a Lévy driven stochastic convolution, also called continuous time Lévy driven moving average model X(t)=∫0ta(t-s)dZ(s)X(t)=int_{0}^{t}a(t-s),dZ(s), where 𝑍 is a Lévy martingale and the kernel a(.)a(,{.},) a deterministic function square integrable on R+mathbb{R}^{+}. Given 𝑁 i.i.d. continuous time observations (Xi(t))t∈[0,T](X_{i}(t))_{tin[0,T]}, i=1,…,Ni=1,dots,N, distributed like (X(t))t∈[0,T](X(t))_{tin[0,T]}, we propose two types of nonparametric projection estimators of a2a^{2} under different sets of assumptions. We bound the L2mathbb{L}^{2}-risk of the estimators and propose a data driven procedure to select the dimension of the projection space, illustrated by a short simulation study.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44550903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I. Almanjahie, S. Bouzebda, Zouaoui Chikr Elmezouar, Ali Laksaci
Abstract The main purpose of the present paper is to investigate the problem of the nonparametric estimation of the expectile regression in which the response variable is scalar while the covariate is a random function. More precisely, an estimator is constructed by using the k Nearest Neighbor procedures (kNN). The main contribution of this study is the establishment of the Uniform consistency in Number of Neighbors (UNN) of the constructed estimator. The usefulness of our result for the smoothing parameter automatic selection is discussed. Short simulation results show that the finite sample performance of the proposed estimator is satisfactory in moderate sample sizes. We finally examine the implementation of this model in practice with a real data in financial risk analysis.
{"title":"The functional kNN estimator of the conditional expectile: Uniform consistency in number of neighbors","authors":"I. Almanjahie, S. Bouzebda, Zouaoui Chikr Elmezouar, Ali Laksaci","doi":"10.1515/strm-2019-0029","DOIUrl":"https://doi.org/10.1515/strm-2019-0029","url":null,"abstract":"Abstract The main purpose of the present paper is to investigate the problem of the nonparametric estimation of the expectile regression in which the response variable is scalar while the covariate is a random function. More precisely, an estimator is constructed by using the k Nearest Neighbor procedures (kNN). The main contribution of this study is the establishment of the Uniform consistency in Number of Neighbors (UNN) of the constructed estimator. The usefulness of our result for the smoothing parameter automatic selection is discussed. Short simulation results show that the finite sample performance of the proposed estimator is satisfactory in moderate sample sizes. We finally examine the implementation of this model in practice with a real data in financial risk analysis.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43495581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the case of a submodular, law-invariant capacity, we provide an entirely elementary proof of a result of Marinacci [M. Marinacci, Upper probabilities and additivity, Sankhyā Ser. A 61 1999, no. 3, 358–361]. As a corollary, we also show that the anticore of a continuous submodular, law-invariant nonatomic capacity has a dichotomous nature: either it is one-dimensional or it is infinite-dimensional. The results have implications for the use of such capacities in financial and economic applications.
摘要在子模、律不变容量的情况下，我们提供了Marinacci结果的一个完全初等的证明[M.Marinacci，Upper probabilities and additivity，SankhyāSer.a 61 1999，no.3358-361]。作为推论，我们还证明了连续子模、律不变的非原子容量的反核具有二分法性质：要么是一维的，要么是无限维的。研究结果对在金融和经济应用中使用这种能力具有启示意义。
{"title":"Bipolar behavior of submodular, law-invariant capacities","authors":"M. Amarante","doi":"10.1515/strm-2020-0025","DOIUrl":"https://doi.org/10.1515/strm-2020-0025","url":null,"abstract":"Abstract In the case of a submodular, law-invariant capacity, we provide an entirely elementary proof of a result of Marinacci [M. Marinacci, Upper probabilities and additivity, Sankhyā Ser. A 61 1999, no. 3, 358–361]. As a corollary, we also show that the anticore of a continuous submodular, law-invariant nonatomic capacity has a dichotomous nature: either it is one-dimensional or it is infinite-dimensional. The results have implications for the use of such capacities in financial and economic applications.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42086447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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