We consider the simulation of a system of decoupled forward-backward stochastic differential equations (FBSDEs) driven by a pure jump Lévy process L and an independent Brownian motion B. We allow the Lévy process L to have an infinite jump activity. Therefore, it is necessary for the simulation to employ a finite approximation of its Lévy measure. We use the generalized shot noise series representation method by Rosiński (2001) to approximate the driving Lévy process L. We compute the Lp error, p ≥ 2, between the true and the approximated FBSDEs which arises from the finite truncation of the shot noise series (given sufficient conditions for existence and uniqueness of the FBSDE). We also derive the Lp error between the true solution and the discretization of the approximated FBSDE using an appropriate backward Euler scheme.
{"title":"Approximation and error analysis of forward-backward SDEs driven by general Lévy processes using shot noise series representations","authors":"Till Massing","doi":"10.1051/ps/2023013","DOIUrl":"https://doi.org/10.1051/ps/2023013","url":null,"abstract":"We consider the simulation of a system of decoupled forward-backward stochastic differential\u0000equations (FBSDEs) driven by a pure jump Lévy process L and an independent Brownian motion\u0000B. We allow the Lévy process L to have an infinite jump activity. Therefore, it is necessary for the\u0000simulation to employ a finite approximation of its Lévy measure. We use the generalized shot noise\u0000series representation method by Rosiński (2001) to approximate the driving Lévy process L. We\u0000compute the Lp error, p ≥ 2, between the true and the approximated FBSDEs which arises from\u0000the finite truncation of the shot noise series (given sufficient conditions for existence and uniqueness\u0000of the FBSDE). We also derive the Lp error between the true solution and the discretization of the\u0000approximated FBSDE using an appropriate backward Euler scheme.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"2015 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86896562","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 investigate a stochastic individual-based model for the population dynamics of host-virus systems where the microbial hosts may transition into a dormant state upon contact with virions, thus evading infection. Such a contact-mediated defence mechanism was described in Bautista&al. (2015) for an archaeal host, while Jackson-Fineran (2019) and Meeske&al. (2019) describe a related, CRISPR-Cas induced, dormancy defense of bacterial hosts to curb phage epidemics. We first analyse the effect of the dormancy-related model parameters on the probability and time of invasion of a newly arriving virus into a resident host population. Given successful invasion, we then show that the emergence (with high probability) of a persistent virus infection (‘epidemic’) in a large host population can be determined by the existence of a coexistence equilibrium for the underlying dynamical system. That is an extension of a dynamical system considered by Beretta-Kuang (1998), known to exhibit a Hopf bifurcation, giving rise to a ‘paradox of enrichment’. We verify that the additional dormancy component can, for certain parameter ranges, prevent the associated loss of stability. Moreover, the presence of contact-mediated dormancy enables the host population to attain higher equilibrium sizes - and still avoid a persistent epidemic - than hosts without this trait.
{"title":"Microbial virus epidemics in the presence of contact-mediated host dormancy","authors":"J. Blath, Andr'as T'obi'as","doi":"10.1051/ps/2022022","DOIUrl":"https://doi.org/10.1051/ps/2022022","url":null,"abstract":"We investigate a stochastic individual-based model for the population dynamics of host-virus systems where the microbial hosts may transition into a dormant state upon contact with virions, thus evading infection. Such a contact-mediated defence mechanism was described in Bautista&al. (2015) for an archaeal host, while Jackson-Fineran (2019) and Meeske&al. (2019) describe a related, CRISPR-Cas induced, dormancy defense of bacterial hosts to curb phage epidemics. We first analyse the effect of the dormancy-related model parameters on the probability and time of invasion of a newly arriving virus into a resident host population. Given successful invasion, we then show that the emergence (with high probability) of a persistent virus infection (‘epidemic’) in a large host population can be determined by the existence of a coexistence equilibrium for the underlying dynamical system. That is an extension of a dynamical system considered by Beretta-Kuang (1998), known to exhibit a Hopf bifurcation, giving rise to a ‘paradox of enrichment’. We verify that the additional dormancy component can, for certain parameter ranges, prevent the associated loss of stability. Moreover, the presence of contact-mediated dormancy enables the host population to attain higher equilibrium sizes - and still avoid a persistent epidemic - than hosts without this trait.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"62 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84728578","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 observe a n-sample, the distribution of which is assumed to belong, or at least to be close enough, to a given mixture model. We propose an estimator of this distribution that belongs to our model and possesses some robustness properties with respect to a possible misspecification of it. We establish a non-asymptotic deviation bound for the Hellinger distance between the target distribution and its estimator when the model consists of a mixture of densities that belong to VC-subgraph classes. Under suitable assumptions and when the mixture model is well-specified, we derive risk bounds for the parameters of the mixture. Finally, we design a statistical procedure that allows us to select from the data the number of components as well as suitable models for each of the densities that are involved in the mixture. These models are chosen among a collection of candidate ones and we show that our selection rule combined with our estimation strategy result in an estimator which satisfies an oracle-type inequality.
{"title":"Robust estimation in finite mixture models","authors":"Alexandre Lecestre","doi":"10.1051/ps/2023004","DOIUrl":"https://doi.org/10.1051/ps/2023004","url":null,"abstract":"We observe a n-sample, the distribution of which is assumed to belong, or at least to be close enough, to a given mixture model. We propose an estimator of this distribution that belongs to our model and possesses some robustness properties with respect to a possible misspecification\u0000of it. We establish a non-asymptotic deviation bound for the Hellinger distance between the target distribution and its estimator when the model consists of a mixture of densities that belong to VC-subgraph classes. Under suitable assumptions and when the mixture model is well-specified, we derive risk bounds for the parameters of the mixture. Finally, we design a statistical procedure that allows us to select from the data the number of components as well as suitable models for each of the densities that are involved in the mixture. These models are chosen among a collection of candidate ones and we show that our selection rule combined with our estimation strategy result in an estimator which satisfies an oracle-type inequality.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"71 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77813724","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}
Via Malliavin calculus, we analyze the limit behavior in distribution of the spatial wavelet variation for the solution to the stochastic linear wave equation with fractional Gaussian noise in time and white noise in space. We propose a wavelet-type estimator for the Hurst parameter of the this solution and we study its asymptotic properties.
{"title":"Wavelet analysis for the solution to the wave equation with fractional noise in time and white noise in space","authors":"Obayda Assaad, C. Tudor","doi":"10.1051/PS/2021009","DOIUrl":"https://doi.org/10.1051/PS/2021009","url":null,"abstract":"Via Malliavin calculus, we analyze the limit behavior in distribution of the spatial wavelet variation for the solution to the stochastic linear wave equation with fractional Gaussian noise in time and white noise in space. We propose a wavelet-type estimator for the Hurst parameter of the this solution and we study its asymptotic properties.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"9 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82140874","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 show that every $R^d$-valued Sobolev path with regularity~$alpha$ and integrability~$p$ can be lifted to a Sobolev rough path provided $1/2 >alpha > 1/p vee 1/3$. The novelty of our approach is its use of ideas underlying Hairer's reconstruction theorem generalized to a framework allowing for Sobolev models and Sobolev modelled distributions. Moreover, we show that the corresponding lifting map is locally Lipschitz continuous with respect to the inhomogeneous Sobolev metric.
我们证明了只要$1/2 > α > 1/p v / 1/3$，具有正则性~$ α $和可积性~$p$的每$R^d$值的Sobolev路径都可以提升为Sobolev粗路径。我们的方法的新颖之处在于它使用了基于harer重建定理的思想，将其推广到一个允许Sobolev模型和Sobolev建模分布的框架。此外，我们还证明了相应的提升映射相对于非齐次Sobolev度规是局部Lipschitz连续的。
{"title":"A Sobolev rough path extension theorem via regularity structures","authors":"Chong Liu, David J. Promel, J. Teichmann","doi":"10.1051/ps/2022016","DOIUrl":"https://doi.org/10.1051/ps/2022016","url":null,"abstract":"We show that every $R^d$-valued Sobolev path with regularity~$alpha$ and integrability~$p$ can be lifted to a Sobolev rough path provided $1/2 >alpha > 1/p vee 1/3$. The novelty of our approach is its use of ideas underlying Hairer's reconstruction theorem generalized to a framework allowing for Sobolev models and Sobolev modelled distributions. Moreover, we show that the corresponding lifting map is locally Lipschitz continuous with respect to the inhomogeneous Sobolev metric.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"15 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88808640","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}
Bipartite networks with exchangeable nodes can be represented by row-column exchangeable matrices. A quadruplet is a submatrix of size 2 x 2. A quadruplet U-statistic is the average of a function on a quadruplet over all the quadruplets of a matrix. We prove several asymptotic results for quadruplet U-statistics on row-column exchangeable matrices, including a weak convergence result in the general case and a central limit theorem when the matrix is also dissociated. These results are applied to statistical inference in network analysis. We suggest a method to perform parameter estimation, network comparison and motifs count for a particular family of row-column exchangeable network models: the bipartite expected degree distribution (BEDD) models. These applications are illustrated by simulations.
{"title":"U-statistics on bipartite exchangeable networks","authors":"T. L. Minh","doi":"10.1051/ps/2023010","DOIUrl":"https://doi.org/10.1051/ps/2023010","url":null,"abstract":"Bipartite networks with exchangeable nodes can be represented by row-column exchangeable matrices. A quadruplet is a submatrix of size 2 x 2. A quadruplet U-statistic is the average of a function on a quadruplet over all the quadruplets of a matrix. We prove several asymptotic results for quadruplet U-statistics on row-column exchangeable matrices, including a weak convergence result in the general case and a central limit theorem when the matrix is also dissociated. These results are applied to statistical inference in network analysis. We suggest a method to perform parameter estimation, network comparison and motifs count for a particular family of row-column exchangeable network models: the bipartite expected degree distribution (BEDD) models. These applications are illustrated by simulations.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"30 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83388378","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 model spatially expanding populations by means of two spatial Λ-Fleming Viot processes (or SLFVs) with selection: the k-parent SLFV and the ∞-parent SLFV. In order to do so, we fill empty areas with type 0 "ghost" individuals with a strong selective disadvantage against "real" type 1 individuals, quantified by a parameter k. The reproduction of ghost individuals is interpreted as local extinction events due to stochasticity in reproduction. When k → +∞, the limiting process, corresponding to the ∞-parent SLFV, is reminiscent of stochastic growth models from percolation theory, but is associated to tools making it possible to investigate the genetic diversity in a population sample. In this article, we provide a rigorous construction of the ∞-parent SLFV, and show that it corresponds to the limit of the k-parent SLFV when k → +∞. In order to do so, we introduce an alternative construction of the k-parent SLFV which allows us to couple SLFVs with different selection strengths and is of interest in its own right. We exhibit three different characterizations of the ∞-parent SLFV, which are valid in different settings and link together population genetics models and stochastic growth models.
{"title":"Stochastic measure-valued models for populations expanding in a continuum","authors":"Apolline Louvet","doi":"10.1051/ps/2022020","DOIUrl":"https://doi.org/10.1051/ps/2022020","url":null,"abstract":"We model spatially expanding populations by means of two spatial Λ-Fleming Viot processes (or SLFVs) with selection: the k-parent SLFV and the ∞-parent SLFV. In order to do so, we fill empty areas with type 0 \"ghost\" individuals with a strong selective disadvantage against \"real\" type 1 individuals, quantified by a parameter k. The reproduction of ghost individuals is interpreted as local extinction events due to stochasticity in reproduction. When k → +∞, the limiting process, corresponding to the ∞-parent SLFV, is reminiscent of stochastic growth models from percolation theory, but is associated to tools making it possible to investigate the genetic diversity in a population sample. In this article, we provide a rigorous construction of the ∞-parent SLFV, and show that it corresponds to the limit of the k-parent SLFV when k → +∞. In order to do so, we introduce an alternative construction of the k-parent SLFV which allows us to couple SLFVs with different selection strengths and is of interest in its own right. We exhibit three different characterizations of the ∞-parent SLFV, which are valid in different settings and link together population genetics models and stochastic growth models.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"133 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78862660","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 study a class of Type-II backward stochastic Volterra integral equations (BSVIEs). For the adapted M-solutions, we obtain two approximation results, namely, a BSDE approximation and a numerical approximation. The BSDE approximation means that the solution of a finite system of backward stochastic differential equations (BSDEs) converges to the adapted M-solution of the original equation. As a consequence of the BSDE approximation, we obtain an estimate for the $L^2$-time regularity of the adapted M-solutions of Type-II BSVIEs. For the numerical approximation, we provide a backward Euler--Maruyama scheme, and show that the scheme converges in the strong $L^2$-sense with the convergence speed of order $1/2$. These results hold true without any differentiability conditions for the coefficients.
{"title":"Approximations for adapted M-solutions of Type-II backward stochastic Volterra integral equations","authors":"Yushi Hamaguchi, Daichi Taguchi","doi":"10.1051/ps/2022017","DOIUrl":"https://doi.org/10.1051/ps/2022017","url":null,"abstract":"In this paper, we study a class of Type-II backward stochastic Volterra integral equations (BSVIEs). For the adapted M-solutions, we obtain two approximation results, namely, a BSDE approximation and a numerical approximation. The BSDE approximation means that the solution of a finite system of backward stochastic differential equations (BSDEs) converges to the adapted M-solution of the original equation. As a consequence of the BSDE approximation, we obtain an estimate for the $L^2$-time regularity of the adapted M-solutions of Type-II BSVIEs. For the numerical approximation, we provide a backward Euler--Maruyama scheme, and show that the scheme converges in the strong $L^2$-sense with the convergence speed of order $1/2$. These results hold true without any differentiability conditions for the coefficients.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"31 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84720029","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 are interested in martingale rearrangement couplings. As introduced by Wiesel in order to prove the stability of Martingale Optimal Transport problems, these are projections in adapted Wasserstein distance of couplings between two probability measures on the real line in the convex order onto the set of martingale couplings between these two marginals. Under a barycentre dispersion assumption on the original coupling which is in particular satisfied by the Hoeffding-Fréchet or comonotone coupling, Wiesel gives a clear algorithmic construction of a martingale rearrangement when the marginals are finitely supported and then gets rid of the finite support assumption by relying on a rather messy limiting procedure to overcome the lack of relative compactness. Here, we give a direct general construction of a martingale rearrangement coupling under the barycentre dispersion assumption. This martingale rearrangement is obtained from the original coupling by an approach similar to the construction we gave in [25] of the inverse transform martingale coupling, a member of a family of martingale couplings close to the Hoeffding-Fréchet coupling, but for a slightly different injection in the set of extended couplings introduced by Beiglböck and Juillet and which involve the uniform distribution on [0,1] in addition to the two marginals.
{"title":"One dimensional martingale rearrangement couplings","authors":"B. Jourdain, W. Margheriti","doi":"10.1051/ps/2022012","DOIUrl":"https://doi.org/10.1051/ps/2022012","url":null,"abstract":"We are interested in martingale rearrangement couplings. As introduced by Wiesel in order to prove the stability of Martingale Optimal Transport problems, these are projections in adapted\u0000\u0000Wasserstein distance of couplings between two probability measures on the real line in the convex order onto the set of martingale couplings between these two marginals. Under a barycentre dispersion assumption on the original coupling which is in particular satisfied by the Hoeffding-Fréchet or comonotone coupling, Wiesel gives a clear algorithmic construction of a martingale rearrangement when the marginals are finitely supported and then gets rid of the finite support assumption by relying on a rather messy limiting procedure to overcome the lack of relative compactness. Here, we give a direct general construction of a martingale rearrangement coupling under the barycentre dispersion assumption. This martingale rearrangement is obtained from the original coupling by an approach similar to the construction we gave in [25] of the inverse transform martingale coupling, a member of a family of martingale couplings close to the Hoeffding-Fréchet coupling, but for a slightly different injection in the set of extended couplings introduced by Beiglböck and Juillet and which involve the uniform distribution on [0,1] in addition to the two marginals.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"6 8 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78504045","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 propose a variable bandwidth kernel regression estimator for i.i.d. observations in ℝ2 to improve the classical Nadaraya-Watson estimator. The bias is improved to the order of O(hn4) under the condition that the fifth order derivative of the density function and the sixth order derivative of the regression function are bounded and continuous. We also establish the central limit theorems for the proposed ideal and true variable kernel regression estimators. The simulation study confirms our results and demonstrates the advantage of the variable bandwidth kernel method over the classical kernel method.
{"title":"Variable bandwidth kernel regression estimation","authors":"Janet Nakarmi, Hailin Sang, Lin Ge","doi":"10.1051/PS/2021003","DOIUrl":"https://doi.org/10.1051/PS/2021003","url":null,"abstract":"In this paper we propose a variable bandwidth kernel regression estimator for i.i.d. observations in ℝ2 to improve the classical Nadaraya-Watson estimator. The bias is improved to the order of O(hn4) under the condition that the fifth order derivative of the density function and the sixth order derivative of the regression function are bounded and continuous. We also establish the central limit theorems for the proposed ideal and true variable kernel regression estimators. The simulation study confirms our results and demonstrates the advantage of the variable bandwidth kernel method over the classical kernel method.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76129130","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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