We define and study properties of implied volatility for American perpetual put options. In particular, we show that if the market prices are derived from a local volatility model with a monotone volatility function, then the corresponding implied volatility is also monotone as a function of the strike price.
{"title":"Monotonicity of implied volatility for perpetual put options","authors":"Erik Ekström, Ebba Mellquist","doi":"10.1017/jpr.2023.36","DOIUrl":"https://doi.org/10.1017/jpr.2023.36","url":null,"abstract":"\u0000 We define and study properties of implied volatility for American perpetual put options. In particular, we show that if the market prices are derived from a local volatility model with a monotone volatility function, then the corresponding implied volatility is also monotone as a function of the strike price.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45725494","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}
Recently, the relevation transformation has received further attention from researchers, and some interesting results have been developed. It is well known that the active redundancy at component level results in a more reliable coherent system than that at system level. However, the lack of study of this problem with relevation redundancy prevents us from fully understanding such a generalization of the active redundancy. In this note we deal with relevation redundancy to coherent systems of homogeneous components. Typically, for a series system of independent components, we have proved that the lifetime of a system with relevation redundancy at component level is larger than that with relevation redundancy at system level in the sense of the usual stochastic order and the likelihood ratio order, respectively. For a coherent system with dependent components, we have developed a sufficient condition in terms of the domination function to the usual stochastic order between the system lifetime with redundancy at component level and that at system level.
{"title":"On Relevation redundancy to coherent systems at component and system levels","authors":"Chen Li, Xiaohui Li","doi":"10.1017/jpr.2023.31","DOIUrl":"https://doi.org/10.1017/jpr.2023.31","url":null,"abstract":"\u0000 Recently, the relevation transformation has received further attention from researchers, and some interesting results have been developed. It is well known that the active redundancy at component level results in a more reliable coherent system than that at system level. However, the lack of study of this problem with relevation redundancy prevents us from fully understanding such a generalization of the active redundancy. In this note we deal with relevation redundancy to coherent systems of homogeneous components. Typically, for a series system of independent components, we have proved that the lifetime of a system with relevation redundancy at component level is larger than that with relevation redundancy at system level in the sense of the usual stochastic order and the likelihood ratio order, respectively. For a coherent system with dependent components, we have developed a sufficient condition in terms of the domination function to the usual stochastic order between the system lifetime with redundancy at component level and that at system level.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44253639","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}
With a focus on the risk contribution in a portofolio of dependent risks, Colini-Baldeschi et al. (2018) introduced Shapley values for variance and standard deviation games. In this note we extend their results, introducing tail variance as well as tail standard deviation games. We derive closed-form expressions for the Shapley values for the tail variance game and we analyze the vector majorization problem for the two games. In particular, we construct two examples showing that the risk contribution rankings for the two games may be inverted depending on the conditioning threshold and the tail fatness. Motivated by these examples, we formulate a conjecture for general portfolios. Lastly, we discuss risk management implications, including the characterization of tail covariance premiums and reinsurance pricing for peer-to-peer insurance policies.
{"title":"Tail variance allocation, Shapley value, and the majorization problem","authors":"M. Galeotti, Giovanni Rabitti","doi":"10.1017/jpr.2023.28","DOIUrl":"https://doi.org/10.1017/jpr.2023.28","url":null,"abstract":"\u0000 With a focus on the risk contribution in a portofolio of dependent risks, Colini-Baldeschi et al. (2018) introduced Shapley values for variance and standard deviation games. In this note we extend their results, introducing tail variance as well as tail standard deviation games. We derive closed-form expressions for the Shapley values for the tail variance game and we analyze the vector majorization problem for the two games. In particular, we construct two examples showing that the risk contribution rankings for the two games may be inverted depending on the conditioning threshold and the tail fatness. Motivated by these examples, we formulate a conjecture for general portfolios. Lastly, we discuss risk management implications, including the characterization of tail covariance premiums and reinsurance pricing for peer-to-peer insurance policies.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44249245","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}
Gaussian process regression is widely used to model an unknown function on a continuous domain by interpolating a discrete set of observed design points. We develop a theoretical framework for proving new moderate deviations inequalities on different types of error probabilities that arise in GP regression. Two specific examples of broad interest are the probability of falsely ordering pairs of points (incorrectly estimating one point as being better than another) and the tail probability of the estimation error at an arbitrary point. Our inequalities connect these probabilities to the mesh norm, which measures how well the design points fill the space.
{"title":"Moderate deviations inequalities for Gaussian process regression","authors":"Jialin Li, I. Ryzhov","doi":"10.1017/jpr.2023.30","DOIUrl":"https://doi.org/10.1017/jpr.2023.30","url":null,"abstract":"\u0000 Gaussian process regression is widely used to model an unknown function on a continuous domain by interpolating a discrete set of observed design points. We develop a theoretical framework for proving new moderate deviations inequalities on different types of error probabilities that arise in GP regression. Two specific examples of broad interest are the probability of falsely ordering pairs of points (incorrectly estimating one point as being better than another) and the tail probability of the estimation error at an arbitrary point. Our inequalities connect these probabilities to the mesh norm, which measures how well the design points fill the space.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47773949","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}
Several information measures have been proposed and studied in the literature. One such measure is extropy, a complementary dual function of entropy. Its meaning and related aging notions have not yet been studied in great detail. In this paper, we first illustrate that extropy information ranks the uniformity of a wide array of absolutely continuous families. We then discuss several theoretical merits of extropy. We also provide a closed-form expression of it for finite mixture distributions. Finally, the dynamic versions of extropy are also discussed, specifically the residual extropy and past extropy measures.
{"title":"Extropy: Characterizations and dynamic versions","authors":"A. Toomaj, M. Hashempour, N. Balakrishnan","doi":"10.1017/jpr.2023.7","DOIUrl":"https://doi.org/10.1017/jpr.2023.7","url":null,"abstract":"\u0000 Several information measures have been proposed and studied in the literature. One such measure is extropy, a complementary dual function of entropy. Its meaning and related aging notions have not yet been studied in great detail. In this paper, we first illustrate that extropy information ranks the uniformity of a wide array of absolutely continuous families. We then discuss several theoretical merits of extropy. We also provide a closed-form expression of it for finite mixture distributions. Finally, the dynamic versions of extropy are also discussed, specifically the residual extropy and past extropy measures.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44805331","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}
Let ${mathrm{d}} X(t) = -Y(t) , {mathrm{d}} t$ , where Y(t) is a one-dimensional diffusion process, and let $tau(x,y)$ be the first time the process (X(t), Y(t)), starting from (x, y), leaves a subset of the first quadrant. The problem of computing the probability $p(x,y),:!=, mathbb{P}[X(tau(x,y))=0]$ is considered. The Laplace transform of the function p(x, y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of $mathbb{E}[tau(x,y)]$ and of the moment-generating function of $tau(x,y)$ can also be derived.
{"title":"A first-passage-place problem for integrated diffusion processes","authors":"M. Lefebvre","doi":"10.1017/jpr.2023.19","DOIUrl":"https://doi.org/10.1017/jpr.2023.19","url":null,"abstract":"\u0000 Let \u0000 \u0000 \u0000 \u0000${mathrm{d}} X(t) = -Y(t) , {mathrm{d}} t$\u0000\u0000 \u0000 , where Y(t) is a one-dimensional diffusion process, and let \u0000 \u0000 \u0000 \u0000$tau(x,y)$\u0000\u0000 \u0000 be the first time the process (X(t), Y(t)), starting from (x, y), leaves a subset of the first quadrant. The problem of computing the probability \u0000 \u0000 \u0000 \u0000$p(x,y),:!=, mathbb{P}[X(tau(x,y))=0]$\u0000\u0000 \u0000 is considered. The Laplace transform of the function p(x, y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of \u0000 \u0000 \u0000 \u0000$mathbb{E}[tau(x,y)]$\u0000\u0000 \u0000 and of the moment-generating function of \u0000 \u0000 \u0000 \u0000$tau(x,y)$\u0000\u0000 \u0000 can also be derived.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49524857","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}
In this paper we consider the problem of averaging for a class of piecewise deterministic Markov processes (PDMPs) whose dynamic is constrained by the presence of a boundary. On reaching the boundary, the process is forced to jump away from it. We assume that this boundary is attractive for the process in question in the sense that its averaged flow is not tangent to it. Our averaging result relies strongly on the existence of densities for the process, allowing us to study the average number of crossings of a smooth hypersurface by an unconstrained PDMP and to deduce from this study averaging results for constrained PDMPs.
{"title":"Averaging for slow–fast piecewise deterministic Markov processes with an attractive boundary","authors":"A. Genadot","doi":"10.1017/jpr.2023.8","DOIUrl":"https://doi.org/10.1017/jpr.2023.8","url":null,"abstract":"\u0000 In this paper we consider the problem of averaging for a class of piecewise deterministic Markov processes (PDMPs) whose dynamic is constrained by the presence of a boundary. On reaching the boundary, the process is forced to jump away from it. We assume that this boundary is attractive for the process in question in the sense that its averaged flow is not tangent to it. Our averaging result relies strongly on the existence of densities for the process, allowing us to study the average number of crossings of a smooth hypersurface by an unconstrained PDMP and to deduce from this study averaging results for constrained PDMPs.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44181111","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 establish new results on the strictly stationary solution to an iterated function system. When the driving sequence is stationary and ergodic, though not independent, the strictly stationary solution may admit no moment but we show an exponential control of the trajectories. We exploit these results to prove, under mild conditions, the consistency of the quasi-maximum likelihood estimator of GARCH(p,q) models with non-independent innovations.
{"title":"Exponential control of the trajectories of iterated function systems with application to semi-strong GARCH models","authors":"Baye Matar Kandji","doi":"10.1017/jpr.2023.13","DOIUrl":"https://doi.org/10.1017/jpr.2023.13","url":null,"abstract":"\u0000 We establish new results on the strictly stationary solution to an iterated function system. When the driving sequence is stationary and ergodic, though not independent, the strictly stationary solution may admit no moment but we show an exponential control of the trajectories. We exploit these results to prove, under mild conditions, the consistency of the quasi-maximum likelihood estimator of GARCH(p,q) models with non-independent innovations.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45204156","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 study the distribution of the consensus formed by a broadcast-based consensus algorithm for cases in which the initial opinions of agents are random variables. We first derive two fundamental equations for the time evolution of the average opinion of agents. Using the derived equations, we then investigate the distribution of the consensus in the limit in which agents do not have any mutual trust, and show that the consensus without mutual trust among agents is in sharp contrast to the consensus with complete mutual trust in the statistical properties if the initial opinion of each agent is integrable. Next, we provide the formulation necessary to mathematically discuss the consensus in the limit in which the number of agents tends to infinity, and derive several results, including a central limit theorem concerning the consensus in this limit. Finally, we study the distribution of the consensus when the initial opinions of agents follow a stable distribution, and show that the consensus also follows a stable distribution in the limit in which the number of agents tends to infinity.
{"title":"Distribution of consensus in a broadcast-based consensus algorithm with random initial opinions","authors":"S. Shioda, Dai Kato","doi":"10.1017/jpr.2023.9","DOIUrl":"https://doi.org/10.1017/jpr.2023.9","url":null,"abstract":"\u0000 We study the distribution of the consensus formed by a broadcast-based consensus algorithm for cases in which the initial opinions of agents are random variables. We first derive two fundamental equations for the time evolution of the average opinion of agents. Using the derived equations, we then investigate the distribution of the consensus in the limit in which agents do not have any mutual trust, and show that the consensus without mutual trust among agents is in sharp contrast to the consensus with complete mutual trust in the statistical properties if the initial opinion of each agent is integrable. Next, we provide the formulation necessary to mathematically discuss the consensus in the limit in which the number of agents tends to infinity, and derive several results, including a central limit theorem concerning the consensus in this limit. Finally, we study the distribution of the consensus when the initial opinions of agents follow a stable distribution, and show that the consensus also follows a stable distribution in the limit in which the number of agents tends to infinity.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43885289","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}
{"title":"JPR volume 60 issue 2 Cover and Front matter","authors":"","doi":"10.1017/jpr.2022.82","DOIUrl":"https://doi.org/10.1017/jpr.2022.82","url":null,"abstract":"","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49434210","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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