The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup,which is defined by the backward differential equation.We provide a proof of the asserti...The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup,which is defined by the backward differential equation.We provide a proof of the assertion of Rhyzhov and Skorokhod(Theory Probab.Appl.,1970)on the uniqueness of the solutions to the equation,which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.展开更多
In this paper we consider general coupled mean-field reflected forward-backward stochastic differential equations(FBSDEs),whose coefficients not only depend on the solution but also on the law of the solution.The firs...In this paper we consider general coupled mean-field reflected forward-backward stochastic differential equations(FBSDEs),whose coefficients not only depend on the solution but also on the law of the solution.The first part of the paper is devoted to the existence and the uniqueness of solutions for such general mean-field reflected backward stochastic differential equations(BSDEs)under Lipschitz conditions,and for the one-dimensional case a comparison theorem is studied.With the help of this comparison result,we prove the existence of the solution for our mean-field reflected forward-backward stochastic differential equation under continuity assumptions.It should be mentioned that,under appropriate assumptions,we prove the uniqueness of this solution as well as that of a comparison theorem for mean-field reflected FBSDEs in a non-trivial manner.展开更多
In this article, we study the multi-dimensional reflected backward stochastic differential equations. The existence and uniqueness result of the solution for this kind of equation is proved by the fixed point argument...In this article, we study the multi-dimensional reflected backward stochastic differential equations. The existence and uniqueness result of the solution for this kind of equation is proved by the fixed point argument where every element of the solution is forced to stay above the given stochastic process, i.e., multi-dimensional obstacle, respectively. We also give a kind of multi-dimensional comparison theorem for the reflected BSDE and then use it as the tool to prove an existence result for the multi-dimensional reflected BSDE where the coefficient is continuous and has linear growth.展开更多
The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of soluti...The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi_linear elliptic type integro_differential equations is obtained.展开更多
The article first studies the fully coupled Forward-Backward Stochastic Differential Equations (FBSDEs) with the continuous local martingale. The article is mainly divided into two parts. In the first part, it consi...The article first studies the fully coupled Forward-Backward Stochastic Differential Equations (FBSDEs) with the continuous local martingale. The article is mainly divided into two parts. In the first part, it considers Backward Stochastic Differential Equations (BSDEs) with the continuous local martingale. Then, on the basis of it, in the second part it considers the fully coupled FBSDEs with the continuous local martingale. It is proved that their solutions exist and are unique under the monotonicity conditions.展开更多
In this paper, we use the solutions of forward-backward stochastic differential equations to get the optimal control for backward stochastic linear quadratic optimal control problem. And we also give the linear feedba...In this paper, we use the solutions of forward-backward stochastic differential equations to get the optimal control for backward stochastic linear quadratic optimal control problem. And we also give the linear feedback regulator for the optimal control problem by using the solutions of a group of Riccati equations.展开更多
In this paper, we propose a new second order numerical scheme for solving backward stochastic differential equations with jumps with the generator linearly depending on . And we theoretically prove that the conv...In this paper, we propose a new second order numerical scheme for solving backward stochastic differential equations with jumps with the generator linearly depending on . And we theoretically prove that the convergence rates of them are of second order for solving and of first order for solving and in norm.展开更多
This paper is concerned with a class of uncertain backward stochastic differential equations (UBSDEs) driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian c...This paper is concerned with a class of uncertain backward stochastic differential equations (UBSDEs) driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties: randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.展开更多
One existence integral condition was obtained for the adapted solution of the general backward stochastic differential equations(BSDEs). Then by solving the integral constraint condition, and using a limit procedure, ...One existence integral condition was obtained for the adapted solution of the general backward stochastic differential equations(BSDEs). Then by solving the integral constraint condition, and using a limit procedure, a new approach method is proposed and the existence of the solution was proved for the BSDEs if the diffusion coefficients satisfy the locally Lipschitz condition. In the special case the solution was a Brownian bridge. The uniqueness is also considered in the meaning of "F0-integrable equivalent class" . The new approach method would give us an efficient way to control the main object instead of the "noise".展开更多
In this paper,a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations(BSDEs).A necessary and sufficient condition is given to judge the L 2-stability of our num...In this paper,a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations(BSDEs).A necessary and sufficient condition is given to judge the L 2-stability of our numerical schemes.This stochastic linear two-step method possesses a family of 3-order convergence schemes in the sense of strong stability.The coefficients in the numerical methods are inferred based on the constraints of strong stability and n-order accuracy(n∈N^(+)).Numerical experiments illustrate that the scheme is an efficient probabilistic numerical method.展开更多
In this work,by combining the multistep discretization in time and the Sinc quadrature rule for approximating the conditional mathematical expectations,we will propose new fully discrete multistep schemes called“Sinc...In this work,by combining the multistep discretization in time and the Sinc quadrature rule for approximating the conditional mathematical expectations,we will propose new fully discrete multistep schemes called“Sinc-multistep schemes”for forward backward stochastic differential equations(FBSDEs).The schemes avoid spatial interpolations and admit high order of convergence.The stability and the K-th order error estimates in time for the K-step Sinc multistep schemes are theoretically proved(1≤K≤6).This seems to be the first time for analyzing fully time-space discrete multistep schemes for FBSDEs.Numerical examples are also presented to demonstrate the effectiveness,stability,and high order of convergence of the proposed schemes.展开更多
In this paper,we study a class of mean-reflected backward doubly stochastic differential equations(MR-BDSDEs),where the constraint depends on the law of the solution and not on its paths.The existence and uniqueness o...In this paper,we study a class of mean-reflected backward doubly stochastic differential equations(MR-BDSDEs),where the constraint depends on the law of the solution and not on its paths.The existence and uniqueness of these solutions were established.The penalization method plays an important role.We also provided a probabilistic interpretation of the classical solutions of the mean-reflected stochastic partial differential equations(MR-SPDEs)in terms of MR-BDSDEs.展开更多
In this paper, we prove that a kind of second order stochastic differential op- erator can be represented by the limit of solutions of BSDEs with uniformly continuous coefficients. This result is a generalization of t...In this paper, we prove that a kind of second order stochastic differential op- erator can be represented by the limit of solutions of BSDEs with uniformly continuous coefficients. This result is a generalization of the representation for the uniformly continuous generator. With the help of this representation, we obtain the corresponding converse comparison theorem for the BSDEs with uniformly continuous coefficients, and get some equivalent relationships between the properties of the generator g and the associated solutions of BSDEs. Moreover, we give a new proof about g-convexity.展开更多
In this paper, the theory of stochastic differential utility is studied. Sufficient conditions for existence, uniqueness, continuity, monotonicity, time consistency, risk aversion and concavity are gived under non-Li...In this paper, the theory of stochastic differential utility is studied. Sufficient conditions for existence, uniqueness, continuity, monotonicity, time consistency, risk aversion and concavity are gived under non-Lipschtz assumptions.展开更多
We study a new algorithm for solvingparabolic partial differential equations(PDEs)and backward stochastic differential equations(BSDEs)in high dimension,which is based on an analogy between the BSDE and reinforcement ...We study a new algorithm for solvingparabolic partial differential equations(PDEs)and backward stochastic differential equations(BSDEs)in high dimension,which is based on an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function,and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE.The policy function is then approximated by a neural network,as is done in deep reinforcement learning.Numerical results using TensorFlow illustrate the efficiency and accuracy of the studied algorithm for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen–Cahn equation,the Hamilton–Jacobi–Bellman equation,and a nonlinear pricing model for financial derivatives.展开更多
In this paper, we deal with a class of one-dimensional backward doubly stochastic differential equations (BDSDEs). We obtain a generalized comparison theorem and a generalized existence theorem of BDSDEs.
In this paper, we deal with one dimensional backward doubly stochastic differential equations (BDSDEs). We obtain a comparison theorem and a uniqueness theorem for BDSDEs with continuous coefficients.
In this work the existence of solutions of one-dimensional backward doubly stochastic differential equations (BDSDEs) with coefficients left-Lipschitz in y (may be discontinuous) and Lipschitz in z is studied. Als...In this work the existence of solutions of one-dimensional backward doubly stochastic differential equations (BDSDEs) with coefficients left-Lipschitz in y (may be discontinuous) and Lipschitz in z is studied. Also, the associated comparison theorem is obtained.展开更多
A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e....A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathe- matical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and (adaptive) hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme.展开更多
This paper considers the fully coupled forward-backward stochastic functional differential equations(FBSFDEs) with stochastic functional differential equations as the forward equations and the generalized anticipated ...This paper considers the fully coupled forward-backward stochastic functional differential equations(FBSFDEs) with stochastic functional differential equations as the forward equations and the generalized anticipated backward stochastic differential equations as the backward equations. The authors will prove the existence and uniqueness theorem for FBSFDEs. As an application, we deal with a quadratic optimal control problem for functional stochastic systems, and get the explicit form of the optimal control by virtue of FBSFDEs.展开更多
基金supported by the National Key R&D Program of China(Grant No.2020YFA0712900)the National Natural Science Foundation of China(Grant No.12271029)。
文摘The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup,which is defined by the backward differential equation.We provide a proof of the assertion of Rhyzhov and Skorokhod(Theory Probab.Appl.,1970)on the uniqueness of the solutions to the equation,which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.
基金supported in part by theNSFC(11871037)Shandong Province(JQ201202)+3 种基金NSFC-RS(11661130148NA150344)111 Project(B12023)supported by the Qingdao Postdoctoral Application Research Project(QDBSH20220202092)。
文摘In this paper we consider general coupled mean-field reflected forward-backward stochastic differential equations(FBSDEs),whose coefficients not only depend on the solution but also on the law of the solution.The first part of the paper is devoted to the existence and the uniqueness of solutions for such general mean-field reflected backward stochastic differential equations(BSDEs)under Lipschitz conditions,and for the one-dimensional case a comparison theorem is studied.With the help of this comparison result,we prove the existence of the solution for our mean-field reflected forward-backward stochastic differential equation under continuity assumptions.It should be mentioned that,under appropriate assumptions,we prove the uniqueness of this solution as well as that of a comparison theorem for mean-field reflected FBSDEs in a non-trivial manner.
基金the National Natural Science Foundation(10371067)the National Basic Research Program of China(973 Program,2007CB814904)+2 种基金the Natural Science Foundation of Shandong Province(Z2006A01)the Doctoral Fund of Education Ministry of China,and Youth Growth Foundation of Shandong University at Weihai, P.R.China. Xiao acknowledges the Natural Science Foundation of Shandong Province (ZR2009AQ017)Independent Innovation Foundation of Shandong University,IIFSDU
文摘In this article, we study the multi-dimensional reflected backward stochastic differential equations. The existence and uniqueness result of the solution for this kind of equation is proved by the fixed point argument where every element of the solution is forced to stay above the given stochastic process, i.e., multi-dimensional obstacle, respectively. We also give a kind of multi-dimensional comparison theorem for the reflected BSDE and then use it as the tool to prove an existence result for the multi-dimensional reflected BSDE where the coefficient is continuous and has linear growth.
文摘The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi_linear elliptic type integro_differential equations is obtained.
文摘The article first studies the fully coupled Forward-Backward Stochastic Differential Equations (FBSDEs) with the continuous local martingale. The article is mainly divided into two parts. In the first part, it considers Backward Stochastic Differential Equations (BSDEs) with the continuous local martingale. Then, on the basis of it, in the second part it considers the fully coupled FBSDEs with the continuous local martingale. It is proved that their solutions exist and are unique under the monotonicity conditions.
基金The NSF(10671112)of ChinaNational Basic Research Program(973 Program)(2007CB814904)of Chinathe NSF(Z2006A01)of Shandong Province and the Chinese New Century Young Teachers Program
文摘In this paper, we use the solutions of forward-backward stochastic differential equations to get the optimal control for backward stochastic linear quadratic optimal control problem. And we also give the linear feedback regulator for the optimal control problem by using the solutions of a group of Riccati equations.
文摘In this paper, we propose a new second order numerical scheme for solving backward stochastic differential equations with jumps with the generator linearly depending on . And we theoretically prove that the convergence rates of them are of second order for solving and of first order for solving and in norm.
基金Supported by National Natural Science Foundation of China(71171003,71210107026)Anhui Natural Science Foundation(10040606003)Anhui Natural Science Foundation of Universities(KJ2012B019,KJ2013B023)
文摘This paper is concerned with a class of uncertain backward stochastic differential equations (UBSDEs) driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties: randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.
基金National Natural Science Foundation of China ( No. 11171062 ) Natural Science Foundation for the Youth,China ( No.11101077) Innovation Program of Shanghai Municipal Education Commission,China ( No. 12ZZ063)
文摘One existence integral condition was obtained for the adapted solution of the general backward stochastic differential equations(BSDEs). Then by solving the integral constraint condition, and using a limit procedure, a new approach method is proposed and the existence of the solution was proved for the BSDEs if the diffusion coefficients satisfy the locally Lipschitz condition. In the special case the solution was a Brownian bridge. The uniqueness is also considered in the meaning of "F0-integrable equivalent class" . The new approach method would give us an efficient way to control the main object instead of the "noise".
基金supported by the National Natural Science of China No.11971263,11871458Shandong Provincial Natural Science Foundation No.ZR2019ZD41National Key R&D Program of China No.2018YFA0703900。
文摘In this paper,a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations(BSDEs).A necessary and sufficient condition is given to judge the L 2-stability of our numerical schemes.This stochastic linear two-step method possesses a family of 3-order convergence schemes in the sense of strong stability.The coefficients in the numerical methods are inferred based on the constraints of strong stability and n-order accuracy(n∈N^(+)).Numerical experiments illustrate that the scheme is an efficient probabilistic numerical method.
基金This work was partially supported by the science challenge project(No.TZ2018001)the national natural science foundation of China(Nos.12071261,11831010 and 11871068)+1 种基金the national key basic research program(No.2018YFA0703900)The authors would like to thank the referees for the helpful comments on the improvement of the present paper.
文摘In this work,by combining the multistep discretization in time and the Sinc quadrature rule for approximating the conditional mathematical expectations,we will propose new fully discrete multistep schemes called“Sinc-multistep schemes”for forward backward stochastic differential equations(FBSDEs).The schemes avoid spatial interpolations and admit high order of convergence.The stability and the K-th order error estimates in time for the K-step Sinc multistep schemes are theoretically proved(1≤K≤6).This seems to be the first time for analyzing fully time-space discrete multistep schemes for FBSDEs.Numerical examples are also presented to demonstrate the effectiveness,stability,and high order of convergence of the proposed schemes.
文摘In this paper,we study a class of mean-reflected backward doubly stochastic differential equations(MR-BDSDEs),where the constraint depends on the law of the solution and not on its paths.The existence and uniqueness of these solutions were established.The penalization method plays an important role.We also provided a probabilistic interpretation of the classical solutions of the mean-reflected stochastic partial differential equations(MR-SPDEs)in terms of MR-BDSDEs.
基金the partial support from the NSF of China(11171186)the NSF of Shandong Province(ZR2010AM021)the "111" project
文摘In this paper, we prove that a kind of second order stochastic differential op- erator can be represented by the limit of solutions of BSDEs with uniformly continuous coefficients. This result is a generalization of the representation for the uniformly continuous generator. With the help of this representation, we obtain the corresponding converse comparison theorem for the BSDEs with uniformly continuous coefficients, and get some equivalent relationships between the properties of the generator g and the associated solutions of BSDEs. Moreover, we give a new proof about g-convexity.
文摘In this paper, the theory of stochastic differential utility is studied. Sufficient conditions for existence, uniqueness, continuity, monotonicity, time consistency, risk aversion and concavity are gived under non-Lipschtz assumptions.
文摘We study a new algorithm for solvingparabolic partial differential equations(PDEs)and backward stochastic differential equations(BSDEs)in high dimension,which is based on an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function,and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE.The policy function is then approximated by a neural network,as is done in deep reinforcement learning.Numerical results using TensorFlow illustrate the efficiency and accuracy of the studied algorithm for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen–Cahn equation,the Hamilton–Jacobi–Bellman equation,and a nonlinear pricing model for financial derivatives.
基金Supported by Marie Curie Initial Training Network (Grant No. PITN-GA2008-213841)National Basic Research Program of China (973 Program, No. 2007CB814906)
文摘In this paper, we deal with a class of one-dimensional backward doubly stochastic differential equations (BDSDEs). We obtain a generalized comparison theorem and a generalized existence theorem of BDSDEs.
基金supported by the Young Scholar Award for Doctoral Students of the Ministry of Education of China, the Marie Curie Initial Training Network(PITN-GA-2008-213841)the National Basic Research Program of China(973 Program,No.2007CB814904)+3 种基金the National Natural Science Foundations of China(No.10921101)Shandong Province(No.2008BS01024)the Science Fund for Distinguished Young Scholars of Shandong Province(No.JQ200801)Shandong University(No.2009JQ004)
文摘In this paper, we deal with one dimensional backward doubly stochastic differential equations (BDSDEs). We obtain a comparison theorem and a uniqueness theorem for BDSDEs with continuous coefficients.
基金Supported by the National Natural Science Foundation of China(Nos.11371226,11071145,11301298,11201268 and 11231005)Foundation for Innovative Research Groups of National Natural Science Foundation of China(No.11221061)+1 种基金the 111 Project(No.B12023)Natural Science Foundation of Shandong Province of China(ZR2012AQ013)
文摘In this work the existence of solutions of one-dimensional backward doubly stochastic differential equations (BDSDEs) with coefficients left-Lipschitz in y (may be discontinuous) and Lipschitz in z is studied. Also, the associated comparison theorem is obtained.
基金Acknowledgments. The first author was supported by the US Air Force Office of Scientific Research under grant FA9550-11-1-0149. The first author was also supported by the Advanced Simulation Computing Research (ASCR), Department of Energy, through the Householder Fellowship at ORNL. The ORNL is operated by UT-Battelle, LLC, for the United States Depart-ment of Energy under Contract DE-AC05-00OR22725. The second author was supported by the US Air Force Office of Scientific Research under grant FA9550-11-1-0149. The third author was supported by the Natural Science Foundation of China under grant 11171189. The third author was also supported by the Natural Science Foundation of China under grant 91130003. The thrid author was also supported by Shandong Province Natural Science Foundation under grant ZR2001AZ002.
文摘A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathe- matical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and (adaptive) hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme.
基金the Program of Natural Science Research of Jiangsu Higher Education Institutions of China under Grant No. 17KJB110009。
文摘This paper considers the fully coupled forward-backward stochastic functional differential equations(FBSFDEs) with stochastic functional differential equations as the forward equations and the generalized anticipated backward stochastic differential equations as the backward equations. The authors will prove the existence and uniqueness theorem for FBSFDEs. As an application, we deal with a quadratic optimal control problem for functional stochastic systems, and get the explicit form of the optimal control by virtue of FBSFDEs.