{"title":"标记霍克斯风险过程的函数近似值","authors":"Laure CoutinIMT, Mahmoud Khabou","doi":"arxiv-2409.06276","DOIUrl":null,"url":null,"abstract":"The marked Hawkes risk process is a compound point process for which the\noccurrence and amplitude of past events impact the future. Thanks to its\nautoregressive properties, it found applications in various fields such as\nneuosciences, social networks and insurance.Since data in real life is acquired\nover a discrete time grid, we propose a strong discrete-time approximation of\nthe continuous-time Hawkes risk process obtained be embedding from the same\nPoisson measure. We then prove trajectorial convergence results both in some\nfractional Sobolev spaces and in the Skorokhod space, hence extending the\ntheorems proven in the literature. We also provide upper bounds on the\nconvergence speed with explicit dependence on the size of the discretisation\nstep, the time horizon and the regularity of the kernel.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"21 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Functional approximation of the marked Hawkes risk process\",\"authors\":\"Laure CoutinIMT, Mahmoud Khabou\",\"doi\":\"arxiv-2409.06276\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The marked Hawkes risk process is a compound point process for which the\\noccurrence and amplitude of past events impact the future. Thanks to its\\nautoregressive properties, it found applications in various fields such as\\nneuosciences, social networks and insurance.Since data in real life is acquired\\nover a discrete time grid, we propose a strong discrete-time approximation of\\nthe continuous-time Hawkes risk process obtained be embedding from the same\\nPoisson measure. We then prove trajectorial convergence results both in some\\nfractional Sobolev spaces and in the Skorokhod space, hence extending the\\ntheorems proven in the literature. We also provide upper bounds on the\\nconvergence speed with explicit dependence on the size of the discretisation\\nstep, the time horizon and the regularity of the kernel.\",\"PeriodicalId\":501245,\"journal\":{\"name\":\"arXiv - MATH - Probability\",\"volume\":\"21 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Probability\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.06276\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06276","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Functional approximation of the marked Hawkes risk process
The marked Hawkes risk process is a compound point process for which the
occurrence and amplitude of past events impact the future. Thanks to its
autoregressive properties, it found applications in various fields such as
neuosciences, social networks and insurance.Since data in real life is acquired
over a discrete time grid, we propose a strong discrete-time approximation of
the continuous-time Hawkes risk process obtained be embedding from the same
Poisson measure. We then prove trajectorial convergence results both in some
fractional Sobolev spaces and in the Skorokhod space, hence extending the
theorems proven in the literature. We also provide upper bounds on the
convergence speed with explicit dependence on the size of the discretisation
step, the time horizon and the regularity of the kernel.