This paper provides sufficient conditions for the time of bankruptcy(of a company or a state)for being a totally inaccessible stopping time and provides the explicit computation of its compensator in a framework where...This paper provides sufficient conditions for the time of bankruptcy(of a company or a state)for being a totally inaccessible stopping time and provides the explicit computation of its compensator in a framework where the flow of market information on the default is modelled explicitly with a Brownian bridge between 0 and 0 on a random time interval.展开更多
In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations （RBSDEs）. We will establish the d...In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations （RBSDEs）. We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bettman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of E1 Karoui, Kapoudjian, Pardoux, Peng and Quenez （1997）, which turns out to be very useful because it allows us to estimate the LP-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.展开更多
We study fully nonlinear second-order(forward)stochastic PDEs.They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework.For the most general fully nonlinear cas...We study fully nonlinear second-order(forward)stochastic PDEs.They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework.For the most general fully nonlinear case,we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions.Our notion of viscosity solutions is equivalent to the alternative using semi-jets.Next,we prove basic properties such as consistency,stability,and a partial comparison principle in the general setting.If the diffusion coefficient is semilinear(i.e,linear in the gradient of the solution and nonlinear in the solution;the drift can still be fully nonlinear),we establish a complete theory,including global existence and a comparison principle.展开更多
Dear All,It is with great pleasure that we welcome you to the first issue of our journal,PUQR–Probability,Uncertainty and Quantitative Risk,a peer-reviewed openaccess journal.Considering its recent and very dynamic d...Dear All,It is with great pleasure that we welcome you to the first issue of our journal,PUQR–Probability,Uncertainty and Quantitative Risk,a peer-reviewed openaccess journal.Considering its recent and very dynamic development,the theory of backward stochastic differential equations has attracted many researchers,with its vast field of applications in stochastic control,games,finance,and deterministic and stochastic partial differential equations.This has spurred the development of new areas for research such as nonlinear dynamic expectation theory,e.g.,g and G-expectation,and path-dependent partial differential equations,while also finding new applications for problems of ambiguity,uncertainty,quantitative risk,and recursive utility in finance and economics.As we further this field,it is important to provide a forum to stimulate future development with a journal that focuses on these topics.More precisely。展开更多
基金supported by the European Community’s FP 7 Program under contract PITN-GA-2008-213841,and Marie Curie ITN《Controlled Systems》.
文摘This paper provides sufficient conditions for the time of bankruptcy(of a company or a state)for being a totally inaccessible stopping time and provides the explicit computation of its compensator in a framework where the flow of market information on the default is modelled explicitly with a Brownian bridge between 0 and 0 on a random time interval.
基金supported by the Agence Nationale de la Recherche (France), reference ANR-10-BLAN 0112the Marie Curie ITN "Controlled Systems", call: FP7-PEOPLE-2007-1-1-ITN, no. 213841-2+3 种基金supported by the National Natural Science Foundation of China (No. 10701050, 11071144)National Basic Research Program of China (973 Program) (No. 2007CB814904)Shandong Province (No. Q2007A04),Independent Innovation Foundation of Shandong Universitythe Project-sponsored by SRF for ROCS, SEM
文摘In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations （RBSDEs）. We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bettman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of E1 Karoui, Kapoudjian, Pardoux, Peng and Quenez （1997）, which turns out to be very useful because it allows us to estimate the LP-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.
文摘We study fully nonlinear second-order(forward)stochastic PDEs.They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework.For the most general fully nonlinear case,we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions.Our notion of viscosity solutions is equivalent to the alternative using semi-jets.Next,we prove basic properties such as consistency,stability,and a partial comparison principle in the general setting.If the diffusion coefficient is semilinear(i.e,linear in the gradient of the solution and nonlinear in the solution;the drift can still be fully nonlinear),we establish a complete theory,including global existence and a comparison principle.
文摘Dear All,It is with great pleasure that we welcome you to the first issue of our journal,PUQR–Probability,Uncertainty and Quantitative Risk,a peer-reviewed openaccess journal.Considering its recent and very dynamic development,the theory of backward stochastic differential equations has attracted many researchers,with its vast field of applications in stochastic control,games,finance,and deterministic and stochastic partial differential equations.This has spurred the development of new areas for research such as nonlinear dynamic expectation theory,e.g.,g and G-expectation,and path-dependent partial differential equations,while also finding new applications for problems of ambiguity,uncertainty,quantitative risk,and recursive utility in finance and economics.As we further this field,it is important to provide a forum to stimulate future development with a journal that focuses on these topics.More precisely。
