Abstract This paper is concerned with a stochastic optimal control problem for a Markov regime switching in the conditional mean field model. Sufficient and necessary maximum principles for optimal control under partial information are obtained. Finally, we illustrate our result through a model which gives an explicit solution.
{"title":"Partial information maximum principle for optimal control problem with regime switching in the conditional mean-field model","authors":"Lazhar Tamer, Hani Ben Abdallah","doi":"10.1515/rose-2022-2094","DOIUrl":"https://doi.org/10.1515/rose-2022-2094","url":null,"abstract":"Abstract This paper is concerned with a stochastic optimal control problem for a Markov regime switching in the conditional mean field model. Sufficient and necessary maximum principles for optimal control under partial information are obtained. Finally, we illustrate our result through a model which gives an explicit solution.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46266769","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 connect the theory of local asymptotic normality (LAN) of Le Cam to Boole’s approximation of the Stratonovich stochastic integral by estimating the parameter in the nonlinear drift coefficient of an ergodic diffusion process satisfying a homogeneous Itô stochastic differential equation based on discretely spaced dense observations of the process. The asymptotic normality and local asymptotic minimaxity (in the Hajek–Le Cam sense) of approximate maximum likelihood estimators, approximate maximum probability estimators and approximate Bayes estimators based on Itô and Boole’s approximations of the continuous likelihood are obtained under an almost slowly increasing experimental design (ASIED) condition ( T n 6 / 7 → 0 {frac{T}{n^{6/7}}to 0} as T → ∞ {Ttoinfty} and n → ∞ {ntoinfty} , where T is the length of the observation time and n is the number of observations) through the weak convergence of the approximate likelihood ratio random fields. Among other things, the Bernstein–von Mises type theorems concerning the convergence of suitably normalized and centered approximate posterior distributions to normal distribution under the same design condition are proved. Asymptotic normality and asymptotic efficiency of the conditional least squares estimator under the same design condition are obtained as a by-product. The log-likelihood derivatives based on Itô approximations are martingales, but the log-likelihood derivatives based on Boole’s approximations are not martingales but weighted averages of forward and backward martingales. These new approximations have faster rate of convergence than the martingale approximations. The methods would have advantages over Euler and Milstein approximations for Monte Carlo simulations.
摘要我们将Le Cam的局部渐近正态性（LAN）理论与Stratonovich随机积分的Boole近似联系起来，通过估计遍历扩散过程的非线性漂移系数中的参数，该过程满足齐次Itô随机微分方程，该过程基于离散间隔的稠密观测。在几乎缓慢增长的实验设计（ASIED）条件下（T n 6/7），获得了基于连续似然的Itô和Boole近似的近似最大似然估计量、近似最大概率估计量和近似贝叶斯估计量的渐近正态性和局部渐近极小性（在Hajek–Le Cam意义上）→ 0｛frac｛T｝｛n^｛6/7｝｝到0｝作为T→ ∞ ｛T to infty｝和n→ ∞ ｛ntoinfty｝，其中T是观测时间的长度，n是观测次数）。证明了Bernstein–von Mises型定理，证明了在相同设计条件下，适当归一化和中心的近似后验分布收敛于正态分布。作为副产品，得到了在相同设计条件下条件最小二乘估计量的渐近正态性和渐近有效性。基于Itô近似的对数似然导数是鞅，但基于Boole近似的对数可能性导数不是鞅，而是前向和后向鞅的加权平均。这些新的近似比鞅近似具有更快的收敛速度。在蒙特卡洛模拟中，该方法将比欧拉和米尔斯坦近似具有优势。
{"title":"Le Cam–Stratonovich–Boole theory for Itô diffusions","authors":"J. Bishwal","doi":"10.1515/rose-2023-2004","DOIUrl":"https://doi.org/10.1515/rose-2023-2004","url":null,"abstract":"Abstract We connect the theory of local asymptotic normality (LAN) of Le Cam to Boole’s approximation of the Stratonovich stochastic integral by estimating the parameter in the nonlinear drift coefficient of an ergodic diffusion process satisfying a homogeneous Itô stochastic differential equation based on discretely spaced dense observations of the process. The asymptotic normality and local asymptotic minimaxity (in the Hajek–Le Cam sense) of approximate maximum likelihood estimators, approximate maximum probability estimators and approximate Bayes estimators based on Itô and Boole’s approximations of the continuous likelihood are obtained under an almost slowly increasing experimental design (ASIED) condition ( T n 6 / 7 → 0 {frac{T}{n^{6/7}}to 0} as T → ∞ {Ttoinfty} and n → ∞ {ntoinfty} , where T is the length of the observation time and n is the number of observations) through the weak convergence of the approximate likelihood ratio random fields. Among other things, the Bernstein–von Mises type theorems concerning the convergence of suitably normalized and centered approximate posterior distributions to normal distribution under the same design condition are proved. Asymptotic normality and asymptotic efficiency of the conditional least squares estimator under the same design condition are obtained as a by-product. The log-likelihood derivatives based on Itô approximations are martingales, but the log-likelihood derivatives based on Boole’s approximations are not martingales but weighted averages of forward and backward martingales. These new approximations have faster rate of convergence than the martingale approximations. The methods would have advantages over Euler and Milstein approximations for Monte Carlo simulations.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49364886","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 Recently, Mao developed a new explicit method, called the truncated Euler–Maruyama method for nonlinear SDEs, and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. The key aim of this paper is to establish the rate of strong convergence of the truncated Euler–Maruyama method for one-dimensional stochastic differential equations involving that the local time at point zero under the drift coefficient satisfies a one-sided Lipschitz condition and plus some additional conditions.
{"title":"The truncated Euler–Maruyama method of one-dimensional stochastic differential equations involving the local time at point zero","authors":"Kamal Hiderah","doi":"10.1515/rose-2023-2003","DOIUrl":"https://doi.org/10.1515/rose-2023-2003","url":null,"abstract":"Abstract Recently, Mao developed a new explicit method, called the truncated Euler–Maruyama method for nonlinear SDEs, and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. The key aim of this paper is to establish the rate of strong convergence of the truncated Euler–Maruyama method for one-dimensional stochastic differential equations involving that the local time at point zero under the drift coefficient satisfies a one-sided Lipschitz condition and plus some additional conditions.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43115526","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 This paper is devoted to study the fractional Pascal noise functionals on compound configuration spaces with special emphasis on the chaotic decomposition of the Hilbert spaces of quadratic integrable functionals with respect to the correlation measure corresponding to the fractional Pascal measure in infinite dimensions.
{"title":"A chaotic decomposition for the fractional Lebesgue–Pascal noise space","authors":"A. Riahi","doi":"10.1515/rose-2023-2005","DOIUrl":"https://doi.org/10.1515/rose-2023-2005","url":null,"abstract":"Abstract This paper is devoted to study the fractional Pascal noise functionals on compound configuration spaces with special emphasis on the chaotic decomposition of the Hilbert spaces of quadratic integrable functionals with respect to the correlation measure corresponding to the fractional Pascal measure in infinite dimensions.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49378668","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 two fractional risk models, where the classical risk process is time-changed by the mixture of tempered stable inverse subordinators. We characterize the risk processes by deriving the marginal distributions and establish the moments and covariance structure. We study the main characteristics of these models such as ruin probability and time to ruin and illustrate the results with Monte Carlo simulations. The data suggest that the ruin time can be approximated by the inverse gaussian distribution and its generalizations.
{"title":"Risk process with mixture of tempered stable inverse subordinators: Analysis and synthesis","authors":"T. Kadankova, Wing Chun Vincent Ng","doi":"10.1515/rose-2022-2096","DOIUrl":"https://doi.org/10.1515/rose-2022-2096","url":null,"abstract":"Abstract We propose two fractional risk models, where the classical risk process is time-changed by the mixture of tempered stable inverse subordinators. We characterize the risk processes by deriving the marginal distributions and establish the moments and covariance structure. We study the main characteristics of these models such as ruin probability and time to ruin and illustrate the results with Monte Carlo simulations. The data suggest that the ruin time can be approximated by the inverse gaussian distribution and its generalizations.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46053245","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}
R. Belfadli, Tarik El Mellali, Imade Fakhouri, Y. Ouknine
Abstract In this paper, we study multidimensional generalized backward stochastic differential equations (GBSDEs), in a general filtration supporting a Brownian motion and an independent Poisson random measure, whose generators are weakly monotone and satisfy a general growth condition with respect to the state variable y. We show that such GBSDEs admit a unique 𝕃 2 {mathbb{L}^{2}} -solution. The main tools and techniques used in the proofs are the a-priori-estimation, the convolution approach, the iteration, the truncation, and the Bihari inequality.
{"title":"𝕃2-solutions of multidimensional generalized BSDEs with weak monotonicity and general growth generators in a general filtration","authors":"R. Belfadli, Tarik El Mellali, Imade Fakhouri, Y. Ouknine","doi":"10.1515/rose-2023-2002","DOIUrl":"https://doi.org/10.1515/rose-2023-2002","url":null,"abstract":"Abstract In this paper, we study multidimensional generalized backward stochastic differential equations (GBSDEs), in a general filtration supporting a Brownian motion and an independent Poisson random measure, whose generators are weakly monotone and satisfy a general growth condition with respect to the state variable y. We show that such GBSDEs admit a unique 𝕃 2 {mathbb{L}^{2}} -solution. The main tools and techniques used in the proofs are the a-priori-estimation, the convolution approach, the iteration, the truncation, and the Bihari inequality.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46753855","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 introduce Wiener integrals with respect to the generalized Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an application, we investigate the corresponding stochastic differential equations with the generalized Hermite process as a driving noise, we prove the existence and the uniqueness of the solution, and we give a generalization of the Hermite Ornstein–Uhlenbeck process and the Hermite-driving Vasicek process.
{"title":"Wiener integrals with respect to the generalized Hermite process (gHp). Applications: SDEs with gHp noise","authors":"Atef Lechiheb","doi":"10.1515/rose-2023-2001","DOIUrl":"https://doi.org/10.1515/rose-2023-2001","url":null,"abstract":"Abstract In this paper, we introduce Wiener integrals with respect to the generalized Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an application, we investigate the corresponding stochastic differential equations with the generalized Hermite process as a driving noise, we prove the existence and the uniqueness of the solution, and we give a generalization of the Hermite Ornstein–Uhlenbeck process and the Hermite-driving Vasicek process.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47341199","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 This paper deals with the problem of existence and uniqueness of 𝕃 p {mathbb{L}^{p}} -solutions for a backward stochastic differential equation in a filtration that supports Lévy processes with p ∈ ( 1 , 2 ) {pin(1,2)} . However, we will focus on when the data satisfy the appropriate integrability conditions and when the coefficient is Lipschitz.
摘要研究了一类支持lsamvy过程(p∈(1,2){pin(1,2)})的滤波中后向随机微分方程的 p {mathbb{L}^{p}}解的存在唯一性问题。然而，我们将重点关注数据何时满足适当的可积条件以及系数何时为Lipschitz。
{"title":"Lp -solution for BSDEs driven by a Lévy process","authors":"M. El Jamali","doi":"10.1515/rose-2023-2006","DOIUrl":"https://doi.org/10.1515/rose-2023-2006","url":null,"abstract":"Abstract This paper deals with the problem of existence and uniqueness of 𝕃 p {mathbb{L}^{p}} -solutions for a backward stochastic differential equation in a filtration that supports Lévy processes with p ∈ ( 1 , 2 ) {pin(1,2)} . However, we will focus on when the data satisfy the appropriate integrability conditions and when the coefficient is Lipschitz.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43627325","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 will study an important property on the regularity of the trajectories of the stochastic flow generated by a famous model in finance. More precisely, we prove the differentiability with respect to initial data of the solution of the stochastic differential equation associated with this model based on Gronwall’s lemma, Itô’s isometry and Burkholder–Davis–Gundy’s and Hölder’s inequalities. This is the main motivation of our research.
{"title":"On the stochastic flow generated by the one default model in one-dimensional case","authors":"Yamina Khatir, Fatima Benziadi, A. Kandouci","doi":"10.1515/rose-2022-2093","DOIUrl":"https://doi.org/10.1515/rose-2022-2093","url":null,"abstract":"Abstract In this paper, we will study an important property on the regularity of the trajectories of the stochastic flow generated by a famous model in finance. More precisely, we prove the differentiability with respect to initial data of the solution of the stochastic differential equation associated with this model based on Gronwall’s lemma, Itô’s isometry and Burkholder–Davis–Gundy’s and Hölder’s inequalities. This is the main motivation of our research.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46925508","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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