In this paper, we focus on anticipated backward stochastic Volterra integral equations(ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytical...In this paper, we focus on anticipated backward stochastic Volterra integral equations(ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps.Motivated by this, a class of dynamic risk measures induced by ABSVIEs with jumps are discussed.展开更多
In this paper, we present a brief survey on the updated theory of backward stochas-tic Volterra integral equations (BSVIEs, for short). BSVIEs are a natural generalization of backward stochastic diff erential equati...In this paper, we present a brief survey on the updated theory of backward stochas-tic Volterra integral equations (BSVIEs, for short). BSVIEs are a natural generalization of backward stochastic diff erential equations (BSDEs, for short). Some interesting motivations of studying BSVIEs are recalled. With proper solution concepts, it is possible to establish the corresponding well-posedness for BSVIEs. We also survey various comparison theorems for solutions to BSVIEs.展开更多
A general type of forward-backward doubly stochastic differential equations (FBDSDEs) is studied. It extends many important equations that have been well studied, including stochastic Hamiltonian systems. Under some...A general type of forward-backward doubly stochastic differential equations (FBDSDEs) is studied. It extends many important equations that have been well studied, including stochastic Hamiltonian systems. Under some much weaker monotonicity assumptions, the existence and uniqueness of measurable solutions are established with a incthod of continuation. Furthermore, the continuity and differentiability of the solutions to FBDSDEs depending on parameters is discussed.展开更多
This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations,where the solution X^(u,ξ)(t)=X(t)is given X(t)=φ(t)+∫_(0)^(t) b(t,s,X(s),u(s))ds+∫_(0)^(t)σ(t,s,X(s...This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations,where the solution X^(u,ξ)(t)=X(t)is given X(t)=φ(t)+∫_(0)^(t) b(t,s,X(s),u(s))ds+∫_(0)^(t)σ(t,s,X(s),u(s))dB(s)+∫_(0)^(t)h(t,s)dξ(s).by Here d B(s)denotes the Brownian motion It?type differential,ξdenotes the singular control(singular in time t with respect to Lebesgue measure)and u denotes the regular control(absolutely continuous with respect to Lebesgue measure).Such systems may for example be used to model harvesting of populations with memory,where X(t)represents the population density at time t,and the singular control processξrepresents the harvesting effort rate.The total income from the harvesting is represented by J(u, ξ) = E[∫_(0)^(t) f_(0)(t,X(t), u(t))dt + ∫_(0)^(t)f_(1)(t,X(t))dξ(t) + g(X(T))] for the given functions f0,f1 and g,where T>0 is a constant denoting the terminal time of the harvesting.Note that it is important to allow the controls to be singular,because in some cases the optimal controls are of this type.Using Hida-Malliavin calculus,we prove sufficient conditions and necessary conditions of optimality of controls.As a consequence,we obtain a new type of backward stochastic Volterra integral equations with singular drift.Finally,to illustrate our results,we apply them to discuss optimal harvesting problems with possibly density dependent prices.展开更多
Backward doubly stochastic differential equations driven by Brownian motions and Poisson process(BDSDEP) with non-Lipschitz coeffcients on random time interval are studied.The probabilistic interpretation for the solu...Backward doubly stochastic differential equations driven by Brownian motions and Poisson process(BDSDEP) with non-Lipschitz coeffcients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations(SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.展开更多
This paper is devoted to the unique solvability of backward stochastic Volterra integral equations (BSVIEs, for short), in terms of both M-solution and the adapted solutions. We prove the existence and uniqueness of...This paper is devoted to the unique solvability of backward stochastic Volterra integral equations (BSVIEs, for short), in terms of both M-solution and the adapted solutions. We prove the existence and uniqueness of M-solutions of BSVIEs in Lp (1 〈 p 〈 2), which extends the existing results on M-solutions. The unique solvability of adapted solutions of BSVIEs in Lp (p 〉 1) is also considered, which also generalizes the results in the existing literature.展开更多
For a backward stochastic differential equation(BSDE,for short),when the generator is not progressively measurable,it might not admit adapted solutions,shown by an example.However,for backward stochastic Volterra inte...For a backward stochastic differential equation(BSDE,for short),when the generator is not progressively measurable,it might not admit adapted solutions,shown by an example.However,for backward stochastic Volterra integral equations(BSVIEs,for short),the generators are allowed to be anticipating.This gives,among other things,an essential difference between BSDEs and BSVIEs.Under some proper conditions,the well-posedness of such BSVIEs is established.Further,the results are extended to path-dependent BSVIEs,in which the generators can depend on the future paths of unknown processes.An additional finding is that for path-dependent BSVIEs,in general,the situation of anticipating generators is not avoidable,and the adaptedness condition similar to that imposed for anticipated BSDEs by Peng−Yang[22]is not necessary.展开更多
We study the existence and uniqueness of a solution to path-dependent backward stochastic Volterra integral equations(BSVIEs)with jumps,where path-dependence means the dependence of the free term and generator of a pa...We study the existence and uniqueness of a solution to path-dependent backward stochastic Volterra integral equations(BSVIEs)with jumps,where path-dependence means the dependence of the free term and generator of a path of a c`adl`ag process.Furthermore,we prove path-differentiability of such a solution and establish the duality principle between a linear path-dependent forward stochastic Volterra integral equation(FSVIE)with jumps and a linear path-dependent BSVIE with jumps.As a result of the duality principle we get a comparison theorem and derive a class of dynamic coherent risk measures based on path-dependent BSVIEs with jumps.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
The notion of bridge is introduced for systems of coupled forward-backward doubly stochastic differential equations (FBDSDEs). It is proved that if two FBDSDEs are linked by a bridge, then they have the same unique so...The notion of bridge is introduced for systems of coupled forward-backward doubly stochastic differential equations (FBDSDEs). It is proved that if two FBDSDEs are linked by a bridge, then they have the same unique solvability. Consequently, by constructing appropriate bridges, we obtain several classes of uniquely solvable FBDSDEs. Finally, the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential equations (SPDEs) combined with algebra equations is given. One distinctive character of this result is that the forward component of the FBDSDEs is coupled with the backward variable.展开更多
In this paper, we study one-dimensional reflected backward doubly stochastic differential equations (RBDSDEs) with one continuous barrier and discontinuous (left or right continuous) genera- tor. We obtain an exis...In this paper, we study one-dimensional reflected backward doubly stochastic differential equations (RBDSDEs) with one continuous barrier and discontinuous (left or right continuous) genera- tor. We obtain an existence theorem and a comparison theorem for solutions of the class of RBDSDEs.展开更多
基金supported by the National Natural Science Foundation of China (11901184, 11771343)the Natural Science Foundation of Hunan Province (2020JJ5025)。
文摘In this paper, we focus on anticipated backward stochastic Volterra integral equations(ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps.Motivated by this, a class of dynamic risk measures induced by ABSVIEs with jumps are discussed.
文摘In this paper, we present a brief survey on the updated theory of backward stochas-tic Volterra integral equations (BSVIEs, for short). BSVIEs are a natural generalization of backward stochastic diff erential equations (BSDEs, for short). Some interesting motivations of studying BSVIEs are recalled. With proper solution concepts, it is possible to establish the corresponding well-posedness for BSVIEs. We also survey various comparison theorems for solutions to BSVIEs.
基金supported by the National Natural Science Foundation of China (No. 10771122)the NaturalScience Foundation of Shandong Province of China (No. Y2006A08)the National Basic ResearchProgram of China (973 Program) (No. 2007CB814900)
文摘A general type of forward-backward doubly stochastic differential equations (FBDSDEs) is studied. It extends many important equations that have been well studied, including stochastic Hamiltonian systems. Under some much weaker monotonicity assumptions, the existence and uniqueness of measurable solutions are established with a incthod of continuation. Furthermore, the continuity and differentiability of the solutions to FBDSDEs depending on parameters is discussed.
基金the financial support provided by the Swedish Research Council grant(2020-04697)the Norwegian Research Council grant(250768/F20),respectively。
文摘This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations,where the solution X^(u,ξ)(t)=X(t)is given X(t)=φ(t)+∫_(0)^(t) b(t,s,X(s),u(s))ds+∫_(0)^(t)σ(t,s,X(s),u(s))dB(s)+∫_(0)^(t)h(t,s)dξ(s).by Here d B(s)denotes the Brownian motion It?type differential,ξdenotes the singular control(singular in time t with respect to Lebesgue measure)and u denotes the regular control(absolutely continuous with respect to Lebesgue measure).Such systems may for example be used to model harvesting of populations with memory,where X(t)represents the population density at time t,and the singular control processξrepresents the harvesting effort rate.The total income from the harvesting is represented by J(u, ξ) = E[∫_(0)^(t) f_(0)(t,X(t), u(t))dt + ∫_(0)^(t)f_(1)(t,X(t))dξ(t) + g(X(T))] for the given functions f0,f1 and g,where T>0 is a constant denoting the terminal time of the harvesting.Note that it is important to allow the controls to be singular,because in some cases the optimal controls are of this type.Using Hida-Malliavin calculus,we prove sufficient conditions and necessary conditions of optimality of controls.As a consequence,we obtain a new type of backward stochastic Volterra integral equations with singular drift.Finally,to illustrate our results,we apply them to discuss optimal harvesting problems with possibly density dependent prices.
基金supported by the National Natural Science Foundation of China (Nos. 10771122,11071145)the Shandong Provincial Natural Science Foundation of China (No. Y2006A08)+2 种基金the Foundation for Innovative Research Groups of National Natural Science Foundation of China (No. 10921101)the National Basic Research Program of China (the 973 Program) (No. 2007CB814900)the Independent Innovation Foundation of Shandong University (No. 2010JQ010)
文摘Backward doubly stochastic differential equations driven by Brownian motions and Poisson process(BDSDEP) with non-Lipschitz coeffcients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations(SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.
基金Supported in part by National Natural Science Foundation of China (Grant Nos. 10771122 and 11071145)Natural Science Foundation of Shandong Province of China (Grant No. Y2006A08)+3 种基金Foundation for Innovative Research Groups of National Natural Science Foundation of China (Grant No. 10921101)National Basic Research Program of China (973 Program, Grant No. 2007CB814900)Independent Innovation Foundation of Shandong University (Grant No. 2010JQ010)Graduate Independent Innovation Foundation of Shandong University (GIIFSDU)
文摘This paper is devoted to the unique solvability of backward stochastic Volterra integral equations (BSVIEs, for short), in terms of both M-solution and the adapted solutions. We prove the existence and uniqueness of M-solutions of BSVIEs in Lp (1 〈 p 〈 2), which extends the existing results on M-solutions. The unique solvability of adapted solutions of BSVIEs in Lp (p 〉 1) is also considered, which also generalizes the results in the existing literature.
基金Hanxiao Wang would like to thank Chenchen Mou(of City University of Hong Kong)for some useful discussionsJiongmin Yong is supported in part by NSF(Grant No.DMS-1812921)+1 种基金Chao Zhou is supported by NSFC(Grant No.11871364)Singapore MOE AcRF(Grant Nos.A-800453-00-00,R-146-000-271-112 and R-146-000-284-114).
文摘For a backward stochastic differential equation(BSDE,for short),when the generator is not progressively measurable,it might not admit adapted solutions,shown by an example.However,for backward stochastic Volterra integral equations(BSVIEs,for short),the generators are allowed to be anticipating.This gives,among other things,an essential difference between BSDEs and BSVIEs.Under some proper conditions,the well-posedness of such BSVIEs is established.Further,the results are extended to path-dependent BSVIEs,in which the generators can depend on the future paths of unknown processes.An additional finding is that for path-dependent BSVIEs,in general,the situation of anticipating generators is not avoidable,and the adaptedness condition similar to that imposed for anticipated BSDEs by Peng−Yang[22]is not necessary.
文摘We study the existence and uniqueness of a solution to path-dependent backward stochastic Volterra integral equations(BSVIEs)with jumps,where path-dependence means the dependence of the free term and generator of a path of a c`adl`ag process.Furthermore,we prove path-differentiability of such a solution and establish the duality principle between a linear path-dependent forward stochastic Volterra integral equation(FSVIE)with jumps and a linear path-dependent BSVIE with jumps.As a result of the duality principle we get a comparison theorem and derive a class of dynamic coherent risk measures based on path-dependent BSVIEs with jumps.
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supported by National Natural Science Foundation of China (Grant Nos. 10771122, 11071145, 10921101 and 11231005)Natural Science Foundation of Shandong Province of China(Grant No. Y2006A08)+1 种基金National Basic Research Program of China (973 Program) (Grant No. 2007CB814900)Independent Innovation Foundation of Shandong University (Grant No. 2010JQ010)
文摘The notion of bridge is introduced for systems of coupled forward-backward doubly stochastic differential equations (FBDSDEs). It is proved that if two FBDSDEs are linked by a bridge, then they have the same unique solvability. Consequently, by constructing appropriate bridges, we obtain several classes of uniquely solvable FBDSDEs. Finally, the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential equations (SPDEs) combined with algebra equations is given. One distinctive character of this result is that the forward component of the FBDSDEs is coupled with the backward variable.
基金Supported by Chinese Natural Science Foundation(Grant No.11271093)the Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20090002110047)
文摘In this paper, we study one-dimensional reflected backward doubly stochastic differential equations (RBDSDEs) with one continuous barrier and discontinuous (left or right continuous) genera- tor. We obtain an existence theorem and a comparison theorem for solutions of the class of RBDSDEs.