This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for$ alpha in(1,2) $, the i.i.d. sequence $ left{ {left( {dfrac{1}{{sqrt n }} displaystylesumlimits_{i = 1}^n {{X_i}} ,dfrac{1}{n} displaystylesumlimits_{i = 1}^n {{Y_i}} ,dfrac{1}{{sqrt[alpha ]{n}}} displaystylesumlimits_{i = 1}^n {{Z_i}} } right)} right}_{n = 1}^infty $converges in distribution to$ tilde{L}_{1} $, where$ tilde{L}_{t}=(tilde {xi}_{t},tilde{eta}_{t},tilde{zeta}_{t}) $,$ tin lbrack0,1] $, is a multidimensional nonlinear Lévy process with an uncertainty set$ Theta $as a set of Lévy triplets. This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation (PIDE) $begin{aligned}[b]left { begin{array} {l} partial_{t}u(t,x,y,z)-sup limits_{(F_{mu},q,Q)in Theta }left { displaystyleint_{mathbb{R}^{d}}delta_{lambda}u(t,x,y,z)F_{mu} ({rm{d}}lambda)right. qquadleft. +langle D_{y}u(t,x,y,z),qrangle+dfrac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]right } =0, u(0,x,y,z)=phi(x,y,z),quad forall(t,x,y,z)in lbrack 0,1]times mathbb{R}^{3d}, end{array} right.end{aligned}$with$ delta_{lambda}u(t,x,y,z):=u(t,x,y,z+lambda)-u(t,x,y,z)-langle D_{z}u(t,x,y,z),lambda rangle $. To construct the limit process$ (tilde {L}_{t})_{tin lbrack0,1]} $, we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space. We further prove a new type of Lévy-Khintchine representation formula to characterize$ (tilde{L}_{t})_{tin lbrack0,1]} $. As a byproduct, we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.
{"title":"A universal robust limit theorem for nonlinear Lévy processes under sublinear expectation","authors":"Mingshang Hu, Lianzi Jiang, Gechun Liang, Shige Peng","doi":"10.3934/puqr.2023001","DOIUrl":"https://doi.org/10.3934/puqr.2023001","url":null,"abstract":"This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for$ alpha in(1,2) $, the i.i.d. sequence $ left{ {left( {dfrac{1}{{sqrt n }} displaystylesumlimits_{i = 1}^n {{X_i}} ,dfrac{1}{n} displaystylesumlimits_{i = 1}^n {{Y_i}} ,dfrac{1}{{sqrt[alpha ]{n}}} displaystylesumlimits_{i = 1}^n {{Z_i}} } right)} right}_{n = 1}^infty $converges in distribution to$ tilde{L}_{1} $, where$ tilde{L}_{t}=(tilde {xi}_{t},tilde{eta}_{t},tilde{zeta}_{t}) $,$ tin lbrack0,1] $, is a multidimensional nonlinear Lévy process with an uncertainty set$ Theta $as a set of Lévy triplets. This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation (PIDE) $begin{aligned}[b]left { begin{array} {l} partial_{t}u(t,x,y,z)-sup limits_{(F_{mu},q,Q)in Theta }left { displaystyleint_{mathbb{R}^{d}}delta_{lambda}u(t,x,y,z)F_{mu} ({rm{d}}lambda)right. qquadleft. +langle D_{y}u(t,x,y,z),qrangle+dfrac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]right } =0, u(0,x,y,z)=phi(x,y,z),quad forall(t,x,y,z)in lbrack 0,1]times mathbb{R}^{3d}, end{array} right.end{aligned}$with$ delta_{lambda}u(t,x,y,z):=u(t,x,y,z+lambda)-u(t,x,y,z)-langle D_{z}u(t,x,y,z),lambda rangle $. To construct the limit process$ (tilde {L}_{t})_{tin lbrack0,1]} $, we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space. We further prove a new type of Lévy-Khintchine representation formula to characterize$ (tilde{L}_{t})_{tin lbrack0,1]} $. As a byproduct, we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134902078","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}
The spirit of now in nowcasting suggests expanding the current to include the near future. Decision theory is then developed by incorporating the consequences of actions into the present. With the future falling into the present discounting it is no longer permitted. Value functions are then observed to be determinate only up to scale and shift that are then locked down by fixing values arbitrarily in two selected states, much like declaring water to freeze and boil at zero and a hundred degrees celsius. The locked down value functions associated policy functions are seen to exist in decision contexts in where the only time is now. Examples are studied in univariate and multivariate dimensions for the decision state space and the dimension of shocks delivering state transitions. The policy functions are expanded from realisitic training sets to the full state space using Gaussian Process Regression. They are implemented on real data with reported performances.
{"title":"Now decision theory","authors":"Dilip B. Madan, Wim Schoutens, King Wang","doi":"10.3934/puqr.2023018","DOIUrl":"https://doi.org/10.3934/puqr.2023018","url":null,"abstract":"The spirit of now in nowcasting suggests expanding the current to include the near future. Decision theory is then developed by incorporating the consequences of actions into the present. With the future falling into the present discounting it is no longer permitted. Value functions are then observed to be determinate only up to scale and shift that are then locked down by fixing values arbitrarily in two selected states, much like declaring water to freeze and boil at zero and a hundred degrees celsius. The locked down value functions associated policy functions are seen to exist in decision contexts in where the only time is now. Examples are studied in univariate and multivariate dimensions for the decision state space and the dimension of shocks delivering state transitions. The policy functions are expanded from realisitic training sets to the full state space using Gaussian Process Regression. They are implemented on real data with reported performances.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135748688","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}
We consider a sequence of independent and identically distributed (i.i.d.) random variables $ {xi_k} $under a sublinear expectation $ mathbb{E} = sup_{PinTheta}E_P $. We first give a new proof to the fact that, under each $ PinTheta $, any cluster point of the empirical averages $ bar{xi}_n = (xi_1+cdots+xi_n)/n $ lies in $ [underline{mu}, overline{mu}] $ with $ underline{mu} = -mathbb{E}[-xi_1], overline{mu} = mathbb{E}[xi_1] $. Next, we consider sublinear expectations on a Polish space $ Omega $, and show that for each constant $ muin [underline{mu},overline{mu}] $, there exists a probability $ P_{mu}inTheta $ such that$ limlimits_{nrightarrow infty}bar{xi}_n = mu, ; P_{mu}text{-a.s.}, $(0.1) supposing that $ Theta $ is weakly compact and $ {xi_n}in L^1_{mathbb{E}}(Omega) $. Under the same conditions, we obtain a generalization of (0.1) in the product space $ Omega = mathbb{R}^{mathbb{N}} $ with $ muin [underline{mu},overline{mu}] $ replaced by $ Pi = pi(xi_1, cdots,xi_d)in [underline{mu},overline{mu}] $. Here $ pi $ is a Borel measurable function on $ mathbb{R}^d $, $ dinmathbb{N} $. Finally, we characterize the triviality of the tail $ sigma $ -algebra of the i.i.d. random variables under a sublinear expectation.
{"title":"A strong law of large numbers under sublinear expectations","authors":"Yongsheng Song","doi":"10.3934/puqr.2023015","DOIUrl":"https://doi.org/10.3934/puqr.2023015","url":null,"abstract":"We consider a sequence of independent and identically distributed (i.i.d.) random variables $ {xi_k} $under a sublinear expectation $ mathbb{E} = sup_{PinTheta}E_P $. We first give a new proof to the fact that, under each $ PinTheta $, any cluster point of the empirical averages $ bar{xi}_n = (xi_1+cdots+xi_n)/n $ lies in $ [underline{mu}, overline{mu}] $ with $ underline{mu} = -mathbb{E}[-xi_1], overline{mu} = mathbb{E}[xi_1] $. Next, we consider sublinear expectations on a Polish space $ Omega $, and show that for each constant $ muin [underline{mu},overline{mu}] $, there exists a probability $ P_{mu}inTheta $ such that$ limlimits_{nrightarrow infty}bar{xi}_n = mu, ; P_{mu}text{-a.s.}, $(0.1) supposing that $ Theta $ is weakly compact and $ {xi_n}in L^1_{mathbb{E}}(Omega) $. Under the same conditions, we obtain a generalization of (0.1) in the product space $ Omega = mathbb{R}^{mathbb{N}} $ with $ muin [underline{mu},overline{mu}] $ replaced by $ Pi = pi(xi_1, cdots,xi_d)in [underline{mu},overline{mu}] $. Here $ pi $ is a Borel measurable function on $ mathbb{R}^d $, $ dinmathbb{N} $. Finally, we characterize the triviality of the tail $ sigma $ -algebra of the i.i.d. random variables under a sublinear expectation.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135748690","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}
We study $ {mathbb{R}}^d $ -valued mean-field stochastic differential equations with a diffusion coefficient that varies in a discontinuous manner on the $ L_p $ -norm of the process. We establish the existence of a unique global strong solution in the presence of a robust drift, while also investigating scenarios where the presence of a global solution is not assured.
{"title":"Mean-field stochastic differential equations with a discontinuous diffusion coefficient","authors":"Jani Nykänen","doi":"10.3934/puqr.2023016","DOIUrl":"https://doi.org/10.3934/puqr.2023016","url":null,"abstract":"We study $ {mathbb{R}}^d $ -valued mean-field stochastic differential equations with a diffusion coefficient that varies in a discontinuous manner on the $ L_p $ -norm of the process. We establish the existence of a unique global strong solution in the presence of a robust drift, while also investigating scenarios where the presence of a global solution is not assured.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135748668","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}
Rui Chen, Roxana Dumitrescu, Andreea Minca, A. Sulem
{"title":"Mean-field BSDEs with jumps and dual representation for global risk measures","authors":"Rui Chen, Roxana Dumitrescu, Andreea Minca, A. Sulem","doi":"10.3934/puqr.2023002","DOIUrl":"https://doi.org/10.3934/puqr.2023002","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"1 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87324037","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}
{"title":"Optimal consumption–investment under partial information in conditionally log-Gaussian models","authors":"H. Nagai","doi":"10.3934/puqr.2023005","DOIUrl":"https://doi.org/10.3934/puqr.2023005","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"58 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91393088","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}
The representation theorem and the viability property for backward stochastic differential equations (BSDEs) require further exploration, given their widespread use in both theory and practical applications. In this study, we present a positive answer to the long-standing open question of whether the representation theorem still holds in the $ L^2 $ -sense under the standard assumptions of square integrability and Lipschitzian continuity on the generators of BSDEs. In the process, the multidimensional case is considered. Subsequently, based on the representation theorem, we obtain a necessary and sufficient condition for the viability property of the BSDEs under standard conditions on the generators. This removes the requirement for the generator to possess the properties of stronger integrability and continuity with respect to time variables. As an application of these results, we conduct various types of comparisons and converse comparisons for the solutions of multidimensional BSDEs, and several properties of the multidimensional $ g $ -expectation are obtained.
后向随机微分方程的表示定理和生存性在理论和实际应用中都有广泛的应用，需要进一步研究。在本研究中，我们给出了一个长期存在的开放性问题，即在平方可积性和Lipschitzian连续性的标准假设下，表示定理在L^2 -意义上是否仍然成立。在此过程中，考虑了多维情况。在此基础上，利用表示定理，得到了在发生器上标准条件下BSDEs生存性的一个充分必要条件。这就消除了对生成器对时间变量具有较强的可积性和连续性的要求。作为这些结果的应用，我们对多维BSDEs的解进行了各种类型的比较和反向比较，得到了多维$ g $ -期望的若干性质。
{"title":"Representation theorem and viability property for multidimensional BSDEs and their applications","authors":"Xuejun Shi, Long Jiang","doi":"10.3934/puqr.2023017","DOIUrl":"https://doi.org/10.3934/puqr.2023017","url":null,"abstract":"The representation theorem and the viability property for backward stochastic differential equations (BSDEs) require further exploration, given their widespread use in both theory and practical applications. In this study, we present a positive answer to the long-standing open question of whether the representation theorem still holds in the $ L^2 $ -sense under the standard assumptions of square integrability and Lipschitzian continuity on the generators of BSDEs. In the process, the multidimensional case is considered. Subsequently, based on the representation theorem, we obtain a necessary and sufficient condition for the viability property of the BSDEs under standard conditions on the generators. This removes the requirement for the generator to possess the properties of stronger integrability and continuity with respect to time variables. As an application of these results, we conduct various types of comparisons and converse comparisons for the solutions of multidimensional BSDEs, and several properties of the multidimensional $ g $ -expectation are obtained.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"280 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135748671","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}
{"title":"BSDEs with stochastic Lipschitz condition: A general result","authors":"Xinying Li, Yu-Chan Lai, Shengjun Fan","doi":"10.3934/puqr.2023011","DOIUrl":"https://doi.org/10.3934/puqr.2023011","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"69 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82073259","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}
In this paper, we study one-dimensional backward stochastic differential equation (BSDE), whose deterministic coefficient $f$ is Lipschitz in $y$ but only continuous in $z$. If the terminal condition $xi$ has bounded Malliavin derivative, we prove some uniqueness results for the BSDE with quadratic and linear growth in $z$, respectively.
{"title":"On the uniqueness result for the BSDE with deterministic coefficient","authors":"Yufeng Shi, Zhi Yang","doi":"10.3934/puqr.2023013","DOIUrl":"https://doi.org/10.3934/puqr.2023013","url":null,"abstract":"In this paper, we study one-dimensional backward stochastic differential equation (BSDE), whose deterministic coefficient $f$ is Lipschitz in $y$ but only continuous in $z$. If the terminal condition $xi$ has bounded Malliavin derivative, we prove some uniqueness results for the BSDE with quadratic and linear growth in $z$, respectively.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135749029","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}
{"title":"3D shear flows driven by Lévy noise at the boundary","authors":"W. Fan, A. Pakzad, Krutika Tawri, R. Temam","doi":"10.3934/puqr.2023004","DOIUrl":"https://doi.org/10.3934/puqr.2023004","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"27 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85349844","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}
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