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Convergence rate of Peng’s law of large numbers under sublinear expectations 次线性期望下彭大数定律的收敛速度
IF 1.5 2区 数学
Probability Uncertainty and Quantitative Risk Pub Date : 2021-07-06 DOI: 10.3934/puqr.2021013
Mingshang Hu, Xiaojuan Li, Xinpeng Li
{"title":"Convergence rate of Peng’s law of large numbers under sublinear expectations","authors":"Mingshang Hu, Xiaojuan Li, Xinpeng Li","doi":"10.3934/puqr.2021013","DOIUrl":"https://doi.org/10.3934/puqr.2021013","url":null,"abstract":"<p style='text-indent:20px;'>This short note provides a new and simple proof of the convergence rate for the Peng’s law of large numbers under sublinear expectations, which improves the results presented by Song [<xref ref-type=\"bibr\" rid=\"b15\">15</xref>] and Fang et al. [<xref ref-type=\"bibr\" rid=\"b3\">3</xref>]. </p>","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81406251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
On the speed of convergence of Picard iterations of backward stochastic differential equations 后向随机微分方程皮卡德迭代的收敛速度
IF 1.5 2区 数学
Probability Uncertainty and Quantitative Risk Pub Date : 2021-07-05 DOI: 10.3934/puqr.2022009
Martin Hutzenthaler, T. Kruse, T. Nguyen
{"title":"On the speed of convergence of Picard iterations of backward stochastic differential equations","authors":"Martin Hutzenthaler, T. Kruse, T. Nguyen","doi":"10.3934/puqr.2022009","DOIUrl":"https://doi.org/10.3934/puqr.2022009","url":null,"abstract":"It is a well-established fact in the scientific literature that Picard iterations of backward stochastic differential equations with globally Lipschitz continuous nonlinearity converge at least exponentially fast to the solution. In this paper we prove that this convergence is in fact at least square-root factorially fast. We show for one example that no higher convergence speed is possible in general. Moreover, if the nonlinearity is z -independent, then the convergence is even factorially fast. Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic differential equations. differential equation, Picard iteration, a priori estimate, semilinear parabolic partial differential equation","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79615857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Stochastic maximum principle for systems driven by local martingales with spatial parameters 空间参数局部鞅驱动系统的随机极大值原理
IF 1.5 2区 数学
Probability Uncertainty and Quantitative Risk Pub Date : 2021-06-02 DOI: 10.3934/puqr.2021011
Jian Song, M. Wang
{"title":"Stochastic maximum principle for systems driven by local martingales with spatial parameters","authors":"Jian Song, M. Wang","doi":"10.3934/puqr.2021011","DOIUrl":"https://doi.org/10.3934/puqr.2021011","url":null,"abstract":"We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter. Assuming the convexity of the control domain, we obtain the stochastic maximum principle as the necessary condition for an optimal control, and we also prove its sufficiency under proper conditions. The stochastic linear quadratic problem in this setting is also discussed.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76019714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Explicit bivariate rate functions for large deviations in AR(1) and MA(1) processes with Gaussian innovations 具有高斯创新的AR(1)和MA(1)过程中大偏差的显式二元速率函数
IF 1.5 2区 数学
Probability Uncertainty and Quantitative Risk Pub Date : 2021-02-18 DOI: 10.3934/puqr.2023008
M. J. Karling, A. Lopes, S. Lopes
{"title":"Explicit bivariate rate functions for large deviations in AR(1) and MA(1) processes with Gaussian innovations","authors":"M. J. Karling, A. Lopes, S. Lopes","doi":"10.3934/puqr.2023008","DOIUrl":"https://doi.org/10.3934/puqr.2023008","url":null,"abstract":"We investigate large deviations properties for centered stationary AR(1) and MA(1) processes with independent Gaussian innovations, by giving the explicit bivariate rate functions for the sequence of random vectors $(boldsymbol{S}_n)_{n in N} = left(n^{-1}(sum_{k=1}^n X_k, sum_{k=1}^n X_k^2)right)_{n in N}$. In the AR(1) case, we also give the explicit rate function for the bivariate random sequence $(W_n)_{n geq 2} = left(n^{-1}(sum_{k=1}^n X_k^2, sum_{k=2}^n X_k X_{k+1})right)_{n geq 2}$. Via Contraction Principle, we provide explicit rate functions for the sequences $(n^{-1} sum_{k=1}^n X_k)_{n in N}$, $(n^{-1} sum_{k=1}^n X_k^2)_{n geq 2}$ and $(n^{-1} sum_{k=2}^n X_k X_{k+1})_{n geq 2}$, as well. In the AR(1) case, we present a new proof for an already known result on the explicit deviation function for the Yule-Walker estimator.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81334768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time 连续时间夏普比率的期限结构与无套利资产定价
IF 1.5 2区 数学
Probability Uncertainty and Quantitative Risk Pub Date : 2021-01-01 DOI: 10.3934/PUQR.2021002
Patrick Beissner, E. R. Gianin
{"title":"The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time","authors":"Patrick Beissner, E. R. Gianin","doi":"10.3934/PUQR.2021002","DOIUrl":"https://doi.org/10.3934/PUQR.2021002","url":null,"abstract":"Motivated by financial and empirical arguments and in order to introduce a more flexible methodology of pricing, we provide a new approach to asset pricing based on Backward Volterra equations. The approach relies on an arbitrage-free and incomplete market setting in continuous time by choosing non-unique pricing measures depending either on the time of evaluation or on the maturity of payoffs. We show that in the latter case the dynamics can be captured by a time-delayed backward stochastic Volterra integral equation here introduced which, to the best of our knowledge, has not yet been studied. We then prove an existence and uniqueness result for time-delayed backward stochastic Volterra integral equations. Finally, we present a Lucas-type consumption-based asset pricing model that justifies the emergence of stochastic discount factors matching the term structure of Sharpe ratios.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73707005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Reduced-form setting under model uncertainty with non-linear affine intensities 具有非线性仿射强度的模型不确定性下的简化形式设置
IF 1.5 2区 数学
Probability Uncertainty and Quantitative Risk Pub Date : 2021-01-01 DOI: 10.3934/puqr.2021008
F. Biagini, Katharina Oberpriller
{"title":"Reduced-form setting under model uncertainty with non-linear affine intensities","authors":"F. Biagini, Katharina Oberpriller","doi":"10.3934/puqr.2021008","DOIUrl":"https://doi.org/10.3934/puqr.2021008","url":null,"abstract":"<p style='text-indent:20px;'>In this paper we extend the reduced-form setting under model uncertainty introduced in [<xref ref-type=\"bibr\" rid=\"b5\">5</xref>] to include intensities following an affine process under parameter uncertainty, as defined in [<xref ref-type=\"bibr\" rid=\"b15\">15</xref>]. This framework allows us to introduce a longevity bond under model uncertainty in a way consistent with the classical case under one prior and to compute its valuation numerically. Moreover, we price a contingent claim with the sublinear conditional operator such that the extended market is still arbitrage-free in the sense of “no arbitrage of the first kind” as in [<xref ref-type=\"bibr\" rid=\"b6\">6</xref>]. </p>","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88347773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Existence, uniqueness and strict comparison theorems for BSDEs driven by RCLL martingales RCLL鞅驱动的BSDEs的存在唯一性及严格比较定理
IF 1.5 2区 数学
Probability Uncertainty and Quantitative Risk Pub Date : 2021-01-01 DOI: 10.3934/puqr.2021016
Tianyang Nie, M. Rutkowski
{"title":"Existence, uniqueness and strict comparison theorems for BSDEs driven by RCLL martingales","authors":"Tianyang Nie, M. Rutkowski","doi":"10.3934/puqr.2021016","DOIUrl":"https://doi.org/10.3934/puqr.2021016","url":null,"abstract":"<p style='text-indent:20px;'>The existence, uniqueness, and strict comparison for solutions to a BSDE driven by a multi-dimensional RCLL martingale are developed. The goal is to develop a general multi-asset framework encompassing a wide spectrum of non-linear financial models with jumps, including as particular cases, the setups studied by Peng and Xu [<xref ref-type=\"bibr\" rid=\"b27\">27</xref>, <xref ref-type=\"bibr\" rid=\"b28\">28</xref>] and Dumitrescu et al. [<xref ref-type=\"bibr\" rid=\"b7\">7</xref>] who dealt with BSDEs driven by a one-dimensional Brownian motion and a purely discontinuous martingale with a single jump. </p>","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90410746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Improved Hoeffding inequality for dependent bounded or sub-Gaussian random variables 依赖有界或亚高斯随机变量的改进Hoeffding不等式
IF 1.5 2区 数学
Probability Uncertainty and Quantitative Risk Pub Date : 2021-01-01 DOI: 10.3934/PUQR.2021003
Y. Tanoue
{"title":"Improved Hoeffding inequality for dependent bounded or sub-Gaussian random variables","authors":"Y. Tanoue","doi":"10.3934/PUQR.2021003","DOIUrl":"https://doi.org/10.3934/PUQR.2021003","url":null,"abstract":"When addressing various financial problems, such as estimating stock portfolio risk, it is necessary to derive the distribution of the sum of the dependent random variables. Although deriving this distribution requires identifying the joint distribution of these random variables, exact estimation of the joint distribution of dependent random variables is difficult. Therefore, in recent years, studies have been conducted on the bound of the sum of dependent random variables with dependence uncertainty. In this study, we obtain an improved Hoeffding inequality for dependent bounded variables. Further, we expand the above result to the case of sub-Gaussian random variables.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90580523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
On the laws of the iterated logarithm under sub-linear expectations 次线性期望下的迭代对数规律
IF 1.5 2区 数学
Probability Uncertainty and Quantitative Risk Pub Date : 2021-01-01 DOI: 10.3934/puqr.2021020
Li-Xin Zhang
{"title":"On the laws of the iterated logarithm under sub-linear expectations","authors":"Li-Xin Zhang","doi":"10.3934/puqr.2021020","DOIUrl":"https://doi.org/10.3934/puqr.2021020","url":null,"abstract":"In this paper, we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space, where the random variables are not necessarily identically distributed. Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm. As an application, the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained. In the paper, it is also shown that if the sub-linear expectation space is rich enough, it will have no continuous capacity. The laws of the iterated logarithm are established without the assumption on the continuity of capacities.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84408573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
CVaR-hedging and its applications to equity-linked life insurance contracts with transaction costs cvar套期保值及其在具有交易成本的股票挂钩人寿保险合同中的应用
IF 1.5 2区 数学
Probability Uncertainty and Quantitative Risk Pub Date : 2021-01-01 DOI: 10.3934/puqr.2021017
Alexander Melnikov,Hongxi Wan
{"title":"CVaR-hedging and its applications to equity-linked life insurance contracts with transaction costs","authors":"Alexander Melnikov,Hongxi Wan","doi":"10.3934/puqr.2021017","DOIUrl":"https://doi.org/10.3934/puqr.2021017","url":null,"abstract":"&lt;p style='text-indent:20px;'&gt;This paper analyzes Conditional Value-at-Risk (CVaR) based partial hedging and its applications on equity-linked life insurance contracts in a Jump-Diffusion market model with transaction costs. A nonlinear partial differential equation (PDE) that an option value process inclusive of transaction costs should satisfy is provided. In particular, the closed-form expression of a European call option price is given. Meanwhile, the CVaR-based partial hedging strategy for a call option is derived explicitly. Both the CVaR hedging price and the weights of the hedging portfolio are based on an adjusted volatility. We obtain estimated values of expected total hedging errors and total transaction costs by a simulation method. Furthermore,our results are implemented to derive target clients’ survival probabilities and age of equity-linked life insurance contracts.&lt;/p&gt;","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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