In this paper,the authors consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way.They assume that the cost function satisfies a c...In this paper,the authors consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way.They assume that the cost function satisfies a convexity and a weak monotonicity property.They use the sufficient Pontryagin principle for optimality to transform the mean field control problem into existence and uniqueness of solution of conditional distribution dependent forward-backward stochastic differential equation(FBSDE for short).They prove the existence and uniqueness of solution of the conditional distribution dependent FBSDE when the dependence of the state on the conditional distribution is sufficiently small,or when the convexity parameter of the running cost on the control is sufficiently large.Two different methods are developed.The first method is based on a continuation of the coefficients,which is developed for FBSDE by[Hu,Y.and Peng,S.,Solution of forward-backward stochastic differential equations,Probab.Theory Rel.,103(2),1995,273–283].They apply the method to conditional distribution dependent FBSDE.The second method is to show the existence result on a small time interval by Banach fixed point theorem and then extend the local solution to the whole time interval.展开更多
In this study,we have analyzed a market impact game between n risk-averse agents who compete for liquidity in a market impact model with a permanent price impact and additional slippage.Most market parameters,includin...In this study,we have analyzed a market impact game between n risk-averse agents who compete for liquidity in a market impact model with a permanent price impact and additional slippage.Most market parameters,including volatility and drift,are allowed to vary stochastically.Our first main result characterizes the Nash equilibrium in terms of a fully coupled system of forward-backward stochastic differential equations(FBSDEs).Our second main result provides conditions under which this system of FBSDEs has a unique solution,resulting in a unique Nash equilibrium.展开更多
In this paper,we consider optimal control of stochastic differential equations subject to an expected path constraint.The stochastic maximum principle is given for a general optimal stochastic control in terms of cons...In this paper,we consider optimal control of stochastic differential equations subject to an expected path constraint.The stochastic maximum principle is given for a general optimal stochastic control in terms of constrained FBSDEs.In particular,the compensated process in our adjoint equation is deterministic,which seems to be new in the literature.For the typical case of linear stochastic systems and quadratic cost functionals(i.e.,the so-called LQ optimal stochastic control),a verification theorem is established,and the existence and uniqueness of the constrained reflected FBSDEs are also given.展开更多
This paper is concerned with linear forward–backward stochastic differential equations(FBSDEs)with state delay,the solvability which is much more complex than the case of no delay or input delay caused by the predict...This paper is concerned with linear forward–backward stochastic differential equations(FBSDEs)with state delay,the solvability which is much more complex than the case of no delay or input delay caused by the prediction of the backward processes of the future time.To overcome this difficulty,we innovatively establish the non-homogeneous relationship between the backward and forward processes with the help of the corresponding discrete-time system.The main contribution is to give the explicit solution to the FBSDEs with state delay in terms of partial Riccati equations for the first time.The presented results form the basis to solve the challenging problem of linear quadratic optimal control for multiplicative-noise stochastic systems with state delay.展开更多
We propose new numerical schemes for decoupled forward-backward stochastic differ- ential equations （FBSDEs） with jumps, where the stochastic dynamics are driven by a d- dimensional Brownian motion and an independen...We propose new numerical schemes for decoupled forward-backward stochastic differ- ential equations （FBSDEs） with jumps, where the stochastic dynamics are driven by a d- dimensional Brownian motion and an independent compensated Poisson random measure. A semi-discrete scheme is developed for discrete time approximation, which is constituted by a classic scheme for the forward SDE [20, 28] and a novel scheme for the backward SDE. Under some reasonable regularity conditions, we prove that the semi-discrete scheme can achieve second-order convergence in approximating the FBSDEs of interest; and such convergence rate does not require jump-adapted temporal discretization. Next, to add in spatial discretization, a fully discrete scheme is developed by designing accurate quadrature rules for estimating the involved conditional mathematical expectations. Several numerical examples are given to illustrate the effectiveness and the high accuracy of the proposed schemes.展开更多
We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations （FBSDEs）. Although the schemes share some common features with the ones ...We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations （FBSDEs）. Although the schemes share some common features with the ones proposed by C. Bender and J. Zhang [Ann. Appl. Probab., 2008, 18： 143-177], less computational work is needed for our method. For both our schemes and the ones proposed by Bender and Zhang, we rigorously obtain first-order error estimates, which improve the half-order error estimates of Bender and Zhang. Moreover, numerical tests are given to demonstrate the first-order accuracy of the schemes.展开更多
This paper focuses on zero-sum stochastic differential games in the framework of forwardbackward stochastic differential equations on a finite time horizon with both players adopting impulse controls.By means of BSDE ...This paper focuses on zero-sum stochastic differential games in the framework of forwardbackward stochastic differential equations on a finite time horizon with both players adopting impulse controls.By means of BSDE methods,in particular that of the notion from Peng’s stochastic backward semigroups,the authors prove a dynamic programming principle for both the upper and the lower value functions of the game.The upper and the lower value functions are then shown to be the unique viscosity solutions of the Hamilton-Jacobi-Bellman-Isaacs equations with a double-obstacle.As a consequence,the uniqueness implies that the upper and lower value functions coincide and the game admits a value.展开更多
This paper concerns a global optimality principle for fully coupled mean-field control systems.Both the first-order and the second-order variational equations are fully coupled mean-field linear FBSDEs. A new linear r...This paper concerns a global optimality principle for fully coupled mean-field control systems.Both the first-order and the second-order variational equations are fully coupled mean-field linear FBSDEs. A new linear relation is introduced, with which we successfully decouple the fully coupled first-order variational equations. We give a new second-order expansion of Y^(ε) that can work well in mean-field framework. Based on this result, the stochastic maximum principle is proved. The comparison with the stochastic maximum principle for controlled mean-field stochastic differential equations is supplied.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
This paper is concerned with the mixed H_2/H_∞ control problem for a new class of stochastic systems with exogenous disturbance signal.The most distinguishing feature,compared with the existing literatures,is that th...This paper is concerned with the mixed H_2/H_∞ control problem for a new class of stochastic systems with exogenous disturbance signal.The most distinguishing feature,compared with the existing literatures,is that the systems are described by linear backward stochastic differential equations(BSDEs).The solution to this problem is obtained completely and explicitly by using an approach which is based primarily on the completion-of-squares technique.Two equivalent expressions for the H_2/H_∞ control are presented.Contrary to forward deterministic and stochastic cases,the solution to the backward stochastic H_2/H_∞ control is no longer feedback of the current state;rather,it is feedback of the entire history of the state.展开更多
基金supported by the National Natural Science Foundation of China(No.12031009)。
文摘In this paper,the authors consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way.They assume that the cost function satisfies a convexity and a weak monotonicity property.They use the sufficient Pontryagin principle for optimality to transform the mean field control problem into existence and uniqueness of solution of conditional distribution dependent forward-backward stochastic differential equation(FBSDE for short).They prove the existence and uniqueness of solution of the conditional distribution dependent FBSDE when the dependence of the state on the conditional distribution is sufficiently small,or when the convexity parameter of the running cost on the control is sufficiently large.Two different methods are developed.The first method is based on a continuation of the coefficients,which is developed for FBSDE by[Hu,Y.and Peng,S.,Solution of forward-backward stochastic differential equations,Probab.Theory Rel.,103(2),1995,273–283].They apply the method to conditional distribution dependent FBSDE.The second method is to show the existence result on a small time interval by Banach fixed point theorem and then extend the local solution to the whole time interval.
基金the National Natural Science Foundation of China(Grant No.11971310)“Assessment of Risk and Uncertainty in Finance”(Grant No.AF0710020)from Shanghai Jiao Tong University+2 种基金Peng Luo gratefully acknowledges the support from the National Natural Science Foundation of China(Grant No.12101400)Peng Luo and Alexander Schied gratefully acknowledge the support from the Natural Sciences and Engineering Research Council of Canada(Grant No.RGPIN-2017-04054)Dewen Xiong gratefully acknowledges the support from the National Natural Science Foundation of China(Grant No.11671257).
文摘In this study,we have analyzed a market impact game between n risk-averse agents who compete for liquidity in a market impact model with a permanent price impact and additional slippage.Most market parameters,including volatility and drift,are allowed to vary stochastically.Our first main result characterizes the Nash equilibrium in terms of a fully coupled system of forward-backward stochastic differential equations(FBSDEs).Our second main result provides conditions under which this system of FBSDEs has a unique solution,resulting in a unique Nash equilibrium.
基金Ying Hu is partially supported by Lebesgue Center of Mathematics“Investissements d’avenir”Program(Grant No.ANR-11-LABX-0020-01)ANR CAESARS(Grant No.ANR-15-CE05-0024)+6 种基金ANR MFG(Grant No.ANR-16-CE40-0015-01)Shanjian Tang is partially supported by the National Science Foundation of China(Grant Nos.11631004 and 12031009)Zuo Quan Xu is partially supported by NSFC(Grant No.11971409)Research Grants Council of Hong Kong(GRF,Grant No.15202421)PolyU-SDU Joint Research Center on Financial MathematicsCAS AMSS-POLYU Joint Laboratory of Applied MathematicsHong Kong Polytechnic University.
文摘In this paper,we consider optimal control of stochastic differential equations subject to an expected path constraint.The stochastic maximum principle is given for a general optimal stochastic control in terms of constrained FBSDEs.In particular,the compensated process in our adjoint equation is deterministic,which seems to be new in the literature.For the typical case of linear stochastic systems and quadratic cost functionals(i.e.,the so-called LQ optimal stochastic control),a verification theorem is established,and the existence and uniqueness of the constrained reflected FBSDEs are also given.
文摘This paper is concerned with linear forward–backward stochastic differential equations(FBSDEs)with state delay,the solvability which is much more complex than the case of no delay or input delay caused by the prediction of the backward processes of the future time.To overcome this difficulty,we innovatively establish the non-homogeneous relationship between the backward and forward processes with the help of the corresponding discrete-time system.The main contribution is to give the explicit solution to the FBSDEs with state delay in terms of partial Riccati equations for the first time.The presented results form the basis to solve the challenging problem of linear quadratic optimal control for multiplicative-noise stochastic systems with state delay.
基金The authors would like to thank the referees for their valuable comments, which improved much of the quality of the paper. This work is partially support- ed by the National Natural Science Foundations of China under grant numbers 91130003,11171189 and 11571206 by Natural Science Foundation of Shandong Province under grant number ZR2011AZ002+1 种基金 by the U.S. Department of Energy, Office of Science, Office of Ad- vanced Scientific Computing Research, Applied Mathematics program under contract number ERKJE45 and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by UT-Battelle, LLC, for the U.S. Department of Energy under Contract DE-AC05-00OR22725.
文摘We propose new numerical schemes for decoupled forward-backward stochastic differ- ential equations （FBSDEs） with jumps, where the stochastic dynamics are driven by a d- dimensional Brownian motion and an independent compensated Poisson random measure. A semi-discrete scheme is developed for discrete time approximation, which is constituted by a classic scheme for the forward SDE [20, 28] and a novel scheme for the backward SDE. Under some reasonable regularity conditions, we prove that the semi-discrete scheme can achieve second-order convergence in approximating the FBSDEs of interest; and such convergence rate does not require jump-adapted temporal discretization. Next, to add in spatial discretization, a fully discrete scheme is developed by designing accurate quadrature rules for estimating the involved conditional mathematical expectations. Several numerical examples are given to illustrate the effectiveness and the high accuracy of the proposed schemes.
基金Acknowledgements The authors would like to thank the referees for the valuable comments, which improved the paper a lot. This work was partially supported by the National Natural Science Foundations of China （Grant Nos. 91130003, 11171189） and the Natural Science Foundation of Shandong Province （No. ZR2011AZ002）.
文摘We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations （FBSDEs）. Although the schemes share some common features with the ones proposed by C. Bender and J. Zhang [Ann. Appl. Probab., 2008, 18： 143-177], less computational work is needed for our method. For both our schemes and the ones proposed by Bender and Zhang, we rigorously obtain first-order error estimates, which improve the half-order error estimates of Bender and Zhang. Moreover, numerical tests are given to demonstrate the first-order accuracy of the schemes.
基金supported by the National Nature Science Foundation of China under Grant Nos.11701040,11871010,61871058the Fundamental Research Funds for the Central Universities under Grant No.2019XDA11。
文摘This paper focuses on zero-sum stochastic differential games in the framework of forwardbackward stochastic differential equations on a finite time horizon with both players adopting impulse controls.By means of BSDE methods,in particular that of the notion from Peng’s stochastic backward semigroups,the authors prove a dynamic programming principle for both the upper and the lower value functions of the game.The upper and the lower value functions are then shown to be the unique viscosity solutions of the Hamilton-Jacobi-Bellman-Isaacs equations with a double-obstacle.As a consequence,the uniqueness implies that the upper and lower value functions coincide and the game admits a value.
基金supported by the Natural Science Foundation of Shandong Province(Grant Nos.ZR2020MA032,ZR2022MA029)National Natural Science Foundation of China(Grant Nos.12171279,72171133).
文摘This paper concerns a global optimality principle for fully coupled mean-field control systems.Both the first-order and the second-order variational equations are fully coupled mean-field linear FBSDEs. A new linear relation is introduced, with which we successfully decouple the fully coupled first-order variational equations. We give a new second-order expansion of Y^(ε) that can work well in mean-field framework. Based on this result, the stochastic maximum principle is proved. The comparison with the stochastic maximum principle for controlled mean-field stochastic differential equations is supplied.
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 the Doctoral Foundation of University of Jinan under Grant No.XBS1213
文摘This paper is concerned with the mixed H_2/H_∞ control problem for a new class of stochastic systems with exogenous disturbance signal.The most distinguishing feature,compared with the existing literatures,is that the systems are described by linear backward stochastic differential equations(BSDEs).The solution to this problem is obtained completely and explicitly by using an approach which is based primarily on the completion-of-squares technique.Two equivalent expressions for the H_2/H_∞ control are presented.Contrary to forward deterministic and stochastic cases,the solution to the backward stochastic H_2/H_∞ control is no longer feedback of the current state;rather,it is feedback of the entire history of the state.
