We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables （t, x） ∈[0, T]× R^d. Thi...We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables （t, x） ∈[0, T]× R^d. This new type of PDEs are formulated through a classical BSDE in which the terminal values and the generators are allowed to be general function of Brownian motion paths. In this way, we establish the nonlinear Feynman- Kac formula for a general non-Markoviau BSDE. Some main properties of solutions of this new PDEs are also obtained.展开更多
The existence and uniqueness of the solutions for a class of backward stochastic differential equations (BSDEs for short) in a random interval are discussed.
A new class of backward particle systems with sequential interaction is proposed to approximate the mean-field backward stochastic differential equations.It is proven that the weighted empirical measure of this partic...A new class of backward particle systems with sequential interaction is proposed to approximate the mean-field backward stochastic differential equations.It is proven that the weighted empirical measure of this particle system converges to the law of the McKean-Vlasov system as the number of particles grows.Based on the Wasserstein met-ric,quantitative propagation of chaos results are obtained for both linear and quadratic growth conditions.Finally,numerical experiments are conducted to validate our theoretical results.展开更多
This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short)with bounded terminal conditions.The generator is allowed to have an unbounded sub-quadratic g...This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short)with bounded terminal conditions.The generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable z.The existence and uniqueness results are given to these BSDEs.As an application,an existence result is given to a system of coupled forward-backward stochastic differential equations with measurable coefficients.展开更多
We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations(BSDEs)with bounded terminal data.By virtue of bounded mean oscillation martingale and change...We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations(BSDEs)with bounded terminal data.By virtue of bounded mean oscillation martingale and change of measure techniques,we obtain stability estimates for the variation of the solutions with different underlying forward processes.In addition,we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs and obtain the explicit rate of convergence by applying the quantitative stability result.展开更多
We consider a McKean Vlasov backward stochastic differential equation(MVBSDE) of the form Y_(t)=-F(t,Y_(t),Z_(t),[Y_(t)]) dt+Z_(t) dB_(t),Y_(T)=ξ,where [Y_(t)] stands for the law of Y,.We show that if F is locally mo...We consider a McKean Vlasov backward stochastic differential equation(MVBSDE) of the form Y_(t)=-F(t,Y_(t),Z_(t),[Y_(t)]) dt+Z_(t) dB_(t),Y_(T)=ξ,where [Y_(t)] stands for the law of Y,.We show that if F is locally monotone in y,locally Lipschitz with respect to z and law's variable,and the monotonicity and Lipschitz constants κ_(n),L_(n) are such that L_(n)^(2)+κ_(n)^(+)=O(log(N)),then the MVBSDE has a unique stable solution.展开更多
The paper investigates the consumption–investment problem for an investor with Epstein–Zin utility in an incomplete market.Closed but not necessarily convex constraints are imposed on strategies.The optimal consumpt...The paper investigates the consumption–investment problem for an investor with Epstein–Zin utility in an incomplete market.Closed but not necessarily convex constraints are imposed on strategies.The optimal consumption and investment strategies are characterized via a quadratic backward stochastic differential equation(BSDE).Due to the stochastic market environment,solutions to this BSDE are unbounded,so the BMO argument breaks down.After establishing the martingale optimality criterion and carefully selecting Lyapunov functions,the verification theorem is ultimately obtained.In addition,several examples and numerical simulations of optimal strategies are provided and illustrated.展开更多
This paper studies the existence and uniqueness of solution of infinite interval backward stochastic differential equation (BSDE) in the plane driven by a Brownian sheet.
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 we study the following nonlinear BSDE：y（t） ＋ ∫1 t f（s,y（s）,z（s））ds ＋ ∫1 t [z（s） ＋ g 1 （s,y（s）） ＋ εg 2 （s,y（s）,z（s））]dW s=ξ,t ∈ [0,1],where ε is a small parameter.The coeffi...In this paper we study the following nonlinear BSDE：y（t） ＋ ∫1 t f（s,y（s）,z（s））ds ＋ ∫1 t [z（s） ＋ g 1 （s,y（s）） ＋ εg 2 （s,y（s）,z（s））]dW s=ξ,t ∈ [0,1],where ε is a small parameter.The coefficient f is locally Lipschitz in y and z,the coefficient g 1 is locally Lipschitz in y,and the coefficient g 2 is uniformly Lipschitz in y and z.Let L N be the locally Lipschitz constant of the coefficients on the ball B（0,N） of R d × R d×r.We prove the existence and uniqueness of the solution when L N ～ √ log N and the parameter ε is small.展开更多
The author studies the optimal investment stopping problem in both continuous and discrete cases, where the investor needs to choose the optimal trading strategy and optimal stopping time concurrently to maximize the ...The author studies the optimal investment stopping problem in both continuous and discrete cases, where the investor needs to choose the optimal trading strategy and optimal stopping time concurrently to maximize the expected utility of terminal wealth.Based on the work of Hu et al.(2018) with an additional stochastic payoff function,the author characterizes the value function for the continuous problem via the theory of quadratic reflected backward stochastic differential equations(BSDEs for short) with unbounded terminal condition. In regard to the discrete problem, she gets the discretization form composed of piecewise quadratic BSDEs recursively under Markovian framework and the assumption of bounded obstacle, and provides some useful a priori estimates about the solutions with the help of an auxiliary forward-backward SDE system and Malliavin calculus. Finally, she obtains the uniform convergence and relevant rate from discretely to continuously quadratic reflected BSDE, which arise from corresponding optimal investment stopping problem through above characterization.展开更多
This paper discusses mean-field backward stochastic differentiM equations （mean-field BS- DEs） with jumps and a new type of controlled mean-field BSDEs with jumps, namely mean-field BSDEs with jumps strongly coupled...This paper discusses mean-field backward stochastic differentiM equations （mean-field BS- DEs） with jumps and a new type of controlled mean-field BSDEs with jumps, namely mean-field BSDEs with jumps strongly coupled with the value function of the associated control problem. The authors first prove the existence and the uniqueness as well as a comparison theorem for the above two types of BSDEs. For this the authors use an approximation method. Then, with the help of the notion of stochastic backward semigroups introduced by Peng in 1997, the authors get the dynamic programming principle （DPP） for the value functions. Furthermore, the authors prove that the value function is a viscosity solution of the associated nonlocal Hamilton-Jacobi-Bellman （HJB） integro-partial differential equation, which is unique in an adequate space of continuous functions introduced by Barles, et al. in 1997.展开更多
基金supported by National Natural Science Foundation of China(Grant No.10921101)the Programme of Introducing Talents of Discipline to Universities of China(Grant No.B12023)the Fundamental Research Funds of Shandong University
文摘We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables （t, x） ∈[0, T]× R^d. This new type of PDEs are formulated through a classical BSDE in which the terminal values and the generators are allowed to be general function of Brownian motion paths. In this way, we establish the nonlinear Feynman- Kac formula for a general non-Markoviau BSDE. Some main properties of solutions of this new PDEs are also obtained.
文摘The existence and uniqueness of the solutions for a class of backward stochastic differential equations (BSDEs for short) in a random interval are discussed.
基金supported by the National Natural Science Foundation of China(No.12222103)the National Key R&D Program of China(No.2018YFA0703900).
文摘A new class of backward particle systems with sequential interaction is proposed to approximate the mean-field backward stochastic differential equations.It is proven that the weighted empirical measure of this particle system converges to the law of the McKean-Vlasov system as the number of particles grows.Based on the Wasserstein met-ric,quantitative propagation of chaos results are obtained for both linear and quadratic growth conditions.Finally,numerical experiments are conducted to validate our theoretical results.
基金supported by the National Natural Science Foundation of China(Nos.11631004,12031009).
文摘This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short)with bounded terminal conditions.The generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable z.The existence and uniqueness results are given to these BSDEs.As an application,an existence result is given to a system of coupled forward-backward stochastic differential equations with measurable coefficients.
基金supported by China Scholarship Council.Gechun Liang is partially supported by the National Natural Science Foundation of China(Grant No.12171169)Guangdong Basic and Applied Basic Research Foundation(Grant No.2019A1515011338)+1 种基金GL thanks J.F.Chassagneux and A.Richou for helpful and inspiring discussions on how to extend to the state dependent volatility case.Shanjian Tang is partially supported by National Science Foundation of China(Grant No.11631004)National Key R&D Program of China(Grant No.2018YFA0703903).
文摘We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations(BSDEs)with bounded terminal data.By virtue of bounded mean oscillation martingale and change of measure techniques,we obtain stability estimates for the variation of the solutions with different underlying forward processes.In addition,we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs and obtain the explicit rate of convergence by applying the quantitative stability result.
文摘We consider a McKean Vlasov backward stochastic differential equation(MVBSDE) of the form Y_(t)=-F(t,Y_(t),Z_(t),[Y_(t)]) dt+Z_(t) dB_(t),Y_(T)=ξ,where [Y_(t)] stands for the law of Y,.We show that if F is locally monotone in y,locally Lipschitz with respect to z and law's variable,and the monotonicity and Lipschitz constants κ_(n),L_(n) are such that L_(n)^(2)+κ_(n)^(+)=O(log(N)),then the MVBSDE has a unique stable solution.
基金supported by the National Natural Science Foundation of China(Grant No.12171471).
文摘The paper investigates the consumption–investment problem for an investor with Epstein–Zin utility in an incomplete market.Closed but not necessarily convex constraints are imposed on strategies.The optimal consumption and investment strategies are characterized via a quadratic backward stochastic differential equation(BSDE).Due to the stochastic market environment,solutions to this BSDE are unbounded,so the BMO argument breaks down.After establishing the martingale optimality criterion and carefully selecting Lyapunov functions,the verification theorem is ultimately obtained.In addition,several examples and numerical simulations of optimal strategies are provided and illustrated.
基金Doctor Promotional Foundation of Shandong Province (Grant 02BS127).
文摘 This paper studies the existence and uniqueness of solution of infinite interval backward stochastic differential equation (BSDE) in the plane driven by a Brownian sheet.
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".
文摘In this paper we study the following nonlinear BSDE：y（t） ＋ ∫1 t f（s,y（s）,z（s））ds ＋ ∫1 t [z（s） ＋ g 1 （s,y（s）） ＋ εg 2 （s,y（s）,z（s））]dW s=ξ,t ∈ [0,1],where ε is a small parameter.The coefficient f is locally Lipschitz in y and z,the coefficient g 1 is locally Lipschitz in y,and the coefficient g 2 is uniformly Lipschitz in y and z.Let L N be the locally Lipschitz constant of the coefficients on the ball B（0,N） of R d × R d×r.We prove the existence and uniqueness of the solution when L N ～ √ log N and the parameter ε is small.
基金This work was supported by the China Scholarship Councilthe National Science Foundation of China(No.11631004)the Science and Technology Commission of Shanghai Municipality(No.14XD1400400)。
文摘The author studies the optimal investment stopping problem in both continuous and discrete cases, where the investor needs to choose the optimal trading strategy and optimal stopping time concurrently to maximize the expected utility of terminal wealth.Based on the work of Hu et al.(2018) with an additional stochastic payoff function,the author characterizes the value function for the continuous problem via the theory of quadratic reflected backward stochastic differential equations(BSDEs for short) with unbounded terminal condition. In regard to the discrete problem, she gets the discretization form composed of piecewise quadratic BSDEs recursively under Markovian framework and the assumption of bounded obstacle, and provides some useful a priori estimates about the solutions with the help of an auxiliary forward-backward SDE system and Malliavin calculus. Finally, she obtains the uniform convergence and relevant rate from discretely to continuously quadratic reflected BSDE, which arise from corresponding optimal investment stopping problem through above characterization.
基金supported by the National Natural Science Foundation of China under Grant Nos.11171187,11222110Shandong Province under Grant No.JQ201202+1 种基金Program for New Century Excellent Talents in University under Grant No.NCET-12-0331111 Project under Grant No.B12023
文摘This paper discusses mean-field backward stochastic differentiM equations （mean-field BS- DEs） with jumps and a new type of controlled mean-field BSDEs with jumps, namely mean-field BSDEs with jumps strongly coupled with the value function of the associated control problem. The authors first prove the existence and the uniqueness as well as a comparison theorem for the above two types of BSDEs. For this the authors use an approximation method. Then, with the help of the notion of stochastic backward semigroups introduced by Peng in 1997, the authors get the dynamic programming principle （DPP） for the value functions. Furthermore, the authors prove that the value function is a viscosity solution of the associated nonlocal Hamilton-Jacobi-Bellman （HJB） integro-partial differential equation, which is unique in an adequate space of continuous functions introduced by Barles, et al. in 1997.
