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## Maximum principle for optimal control of fully coupled forwardbackward stochastic differential delayed equations

Venue: | ESAIM: Control, Optimisation and Calculus of Variations |

Citations: | 1 - 1 self |

### Citations

538 | Backward stochastic differential equations in finance
- Karoui, Peng, et al.
- 1997
(Show Context)
Citation Context ...SDDEs. General nonlinear BSDEs were developed by Pardoux and Peng [16] and have been widely applied in optimal control, finance and partial differential equations (see Peng [17, 18], El Karoui et al. =-=[10]-=-, Yong and Zhou [27]). 1076 J. HUANG AND J. SHI Recently, Chen and Wu [5] studied one kind of delayed stochastic optimal control problem. When introducing the adjoint equation, they encountered some n... |

468 |
Adapted solution of a backward stochastic differential equation
- Pardoux, Peng
- 1990
(Show Context)
Citation Context ... their past history. For example, the evolution of the stock price and other stochastic dynamical systems are sometimes identified as SDDEs. General nonlinear BSDEs were developed by Pardoux and Peng =-=[16]-=- and have been widely applied in optimal control, finance and partial differential equations (see Peng [17, 18], El Karoui et al. [10], Yong and Zhou [27]). 1076 J. HUANG AND J. SHI Recently, Chen and... |

219 |
Stochastic controls: Hamiltonian Systems and HJB equations
- Yong, Zhou
- 1999
(Show Context)
Citation Context ...near BSDEs were developed by Pardoux and Peng [16] and have been widely applied in optimal control, finance and partial differential equations (see Peng [17, 18], El Karoui et al. [10], Yong and Zhou =-=[27]-=-). 1076 J. HUANG AND J. SHI Recently, Chen and Wu [5] studied one kind of delayed stochastic optimal control problem. When introducing the adjoint equation, they encountered some new type of BSDEs. Th... |

85 |
Backward-forward stochastic differential equations
- Antonelli
- 1993
(Show Context)
Citation Context ...tochastic differential equations (FBSDEs) are widely used in mathematical economics and mathematical finance. They are encountered in stochastic recursive utility optimization problems (see Antonelli =-=[1]-=-, El Karoui et al. [10], Wang and Wu [23]) and principal-agent problems (see Williams [24], Cvitanic et al. [9]). Moreover, some financial optimization problems for large investors (see Cvitanic and M... |

64 |
Hedging options for a large investor and forward–backward SDEs.
- Cvitanic, Ma
- 1996
(Show Context)
Citation Context ...El Karoui et al. [10], Wang and Wu [23]) and principal-agent problems (see Williams [24], Cvitanic et al. [9]). Moreover, some financial optimization problems for large investors (see Cvitanic and Ma =-=[8]-=-, Cucoo and Cvitanic [7], Buckdahn and Hu [3]) and some asset pricing problems with forward-backward differential utility (see Antonelli [1], Antonelli et al. [2]) will directly lead to fully coupled ... |

60 |
Solution of forward-backward stochastic differential equations
- Hu, Peng
(Show Context)
Citation Context ...the solution of such AFBSDDE under the above G-monotonic assumptions has been studied recently by Chen and Wu [6] (noting that this kind G-monotonic assumption was initially introduced by Hu and Peng =-=[11]-=-, Peng and Wu [19]). Lemma 1.1 ([6]). Let (H1) and (H2)( or (H2)’) hold. Then for any v(·) ∈ Uad, AFBSDDE (1.1) admits a unique adapted solution (xv(·), yv(·), zv(·)) ∈ L2F([−δ, T ];Rn)× L2F([0, T + δ... |

50 | Fully coupled forward-backward stochastic differential equations and applications to optimal control
- Peng, Wu
- 1999
(Show Context)
Citation Context ...ch AFBSDDE under the above G-monotonic assumptions has been studied recently by Chen and Wu [6] (noting that this kind G-monotonic assumption was initially introduced by Hu and Peng [11], Peng and Wu =-=[19]-=-). Lemma 1.1 ([6]). Let (H1) and (H2)( or (H2)’) hold. Then for any v(·) ∈ Uad, AFBSDDE (1.1) admits a unique adapted solution (xv(·), yv(·), zv(·)) ∈ L2F([−δ, T ];Rn)× L2F([0, T + δ];Rm)× L2F([0, T ]... |

36 |
Optimal consumption choices for a ‘large’ investor,”
- Cuoco, Cvitanic
- 1998
(Show Context)
Citation Context ...ang and Wu [23]) and principal-agent problems (see Williams [24], Cvitanic et al. [9]). Moreover, some financial optimization problems for large investors (see Cvitanic and Ma [8], Cucoo and Cvitanic =-=[7]-=-, Buckdahn and Hu [3]) and some asset pricing problems with forward-backward differential utility (see Antonelli [1], Antonelli et al. [2]) will directly lead to fully coupled FBSDEs. So the optimal c... |

29 | On Dynamic Principal-Agent Problems in Continuous Time. Working paper,
- Williams
- 2009
(Show Context)
Citation Context ...athematical finance. They are encountered in stochastic recursive utility optimization problems (see Antonelli [1], El Karoui et al. [10], Wang and Wu [23]) and principal-agent problems (see Williams =-=[24]-=-, Cvitanic et al. [9]). Moreover, some financial optimization problems for large investors (see Cvitanic and Ma [8], Cucoo and Cvitanic [7], Buckdahn and Hu [3]) and some asset pricing problems with f... |

18 |
A maximum principle for optimal control of stochastic systems with delay, with applications to finance
- Øksendal, Sulem
- 2001
(Show Context)
Citation Context ...s convex in x, γ is convex in y. The optimal control problem for stochastic differential delayed systems has been studied by many researchers (see Kolmanovskii and Maizenberg [12], Øksendal and Sulem =-=[15]-=-, Chen and Wu [5] and the references therein) which has the practical background. One of the main motivations is that many random phenomena have the feature of past-dependence, i.e., their behavior at... |

14 | Anticipated backward stochastic differential equations
- Peng, Yang
(Show Context)
Citation Context ...rt with terminal condition is an anticipated backward stochastic differential equation (ABSDE). The general theory and applications of SDDEs and ABSDEs can be found in Mohammed [14] and Peng and Yang =-=[20]-=-, respectively. In addition, one distinguished feature of equation (1.1) is that the forward SDDE and backward ABSDE are fully coupled. The existence and uniqueness of the solution of such AFBSDDE und... |

13 |
The maximum principle for fully coupled forwardbackward stochastic control system
- Shi, Wu
(Show Context)
Citation Context ... necessarily convex and the control system described by the fully coupled AFBSDDE, this problem is more difficult. We overcome the difficulties by techniques dealing with fully-coupling in Shi and Wu =-=[21]-=-, by methods dealing with time delay in Chen and Wu [5] and by Clarke generalized gradient’s approach dealing with sufficient conditions of optimality in Zhou [30]. We refer to Wu [25], Shi and Wu [21... |

11 |
Hedging contingent claims for a large investor in an incomplete market
- Buckdahn, Hu
- 1998
(Show Context)
Citation Context ...principal-agent problems (see Williams [24], Cvitanic et al. [9]). Moreover, some financial optimization problems for large investors (see Cvitanic and Ma [8], Cucoo and Cvitanic [7], Buckdahn and Hu =-=[3]-=-) and some asset pricing problems with forward-backward differential utility (see Antonelli [1], Antonelli et al. [2]) will directly lead to fully coupled FBSDEs. So the optimal control problems for f... |

11 |
Optimal control of stochastic systems with aftereffect, Autom
- Kolmanovskii, Maizenberg
- 1973
(Show Context)
Citation Context ..., MT ∈ Rm×n, ∀x ∈ Rn; Φ is convex in x, γ is convex in y. The optimal control problem for stochastic differential delayed systems has been studied by many researchers (see Kolmanovskii and Maizenberg =-=[12]-=-, Øksendal and Sulem [15], Chen and Wu [5] and the references therein) which has the practical background. One of the main motivations is that many random phenomena have the feature of past-dependence... |

11 |
Stochastic Differential Equations with Memory: Theory, Examples and Applications
- Mohammed
- 1996
(Show Context)
Citation Context ...DE) and the backward part with terminal condition is an anticipated backward stochastic differential equation (ABSDE). The general theory and applications of SDDEs and ABSDEs can be found in Mohammed =-=[14]-=- and Peng and Yang [20], respectively. In addition, one distinguished feature of equation (1.1) is that the forward SDDE and backward ABSDE are fully coupled. The existence and uniqueness of the solut... |

10 |
Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems
- Wu
- 1998
(Show Context)
Citation Context ...g in Shi and Wu [21], by methods dealing with time delay in Chen and Wu [5] and by Clarke generalized gradient’s approach dealing with sufficient conditions of optimality in Zhou [30]. We refer to Wu =-=[25]-=-, Shi and Wu [21, 22], Meng [13], Yong [26] for more details on maximum principles for fully coupled forward-backward stochastic systems without delay. The rest of this paper is organized as follows. ... |

8 |
Maximum principle for the stochastic optimal control problem with delay and application
- Chen, Wu
- 2010
(Show Context)
Citation Context ...hastic systems are extensively studied in the literature. However, to our best knowledge, there are few papers studying forward-backward stochastic differential delayed systems. Recently, Chen and Wu =-=[4]-=- discussed one kind of stochastic recursive optimal control problem of the system described by FBSDE with time-varying delay. The necessary condition for the optimal control – maximum principle – is d... |

8 |
Optimality Variational Principle for Controlled ForwardBackward Stochastic Differential Equations with Mixed InitialTerminal Conditions
- Yong
- 2010
(Show Context)
Citation Context ...th time delay in Chen and Wu [5] and by Clarke generalized gradient’s approach dealing with sufficient conditions of optimality in Zhou [30]. We refer to Wu [25], Shi and Wu [21, 22], Meng [13], Yong =-=[26]-=- for more details on maximum principles for fully coupled forward-backward stochastic systems without delay. The rest of this paper is organized as follows. In Section 2, we give the main results of t... |

7 |
Asset pricing with a forwardbackward stochastic differential utility
- Antonelli, Barucci, et al.
(Show Context)
Citation Context ... large investors (see Cvitanic and Ma [8], Cucoo and Cvitanic [7], Buckdahn and Hu [3]) and some asset pricing problems with forward-backward differential utility (see Antonelli [1], Antonelli et al. =-=[2]-=-) will directly lead to fully coupled FBSDEs. So the optimal control problems for forwardbackward stochastic systems are extensively studied in the literature. However, to our best knowledge, there ar... |

6 |
The maximum principle for stochastic recursive optimal control problems under partial information
- Wang, Wu
(Show Context)
Citation Context ...) are widely used in mathematical economics and mathematical finance. They are encountered in stochastic recursive utility optimization problems (see Antonelli [1], El Karoui et al. [10], Wang and Wu =-=[23]-=-) and principal-agent problems (see Williams [24], Cvitanic et al. [9]). Moreover, some financial optimization problems for large investors (see Cvitanic and Ma [8], Cucoo and Cvitanic [7], Buckdahn a... |

6 |
Sufficient conditions of optimality for stochastic systems with controllable diffusions
- Zhou
(Show Context)
Citation Context ...ng with fully-coupling in Shi and Wu [21], by methods dealing with time delay in Chen and Wu [5] and by Clarke generalized gradient’s approach dealing with sufficient conditions of optimality in Zhou =-=[30]-=-. We refer to Wu [25], Shi and Wu [21, 22], Meng [13], Yong [26] for more details on maximum principles for fully coupled forward-backward stochastic systems without delay. The rest of this paper is o... |

5 | Optimal contracts in continuous-time models
- Cvitanic, Wan, et al.
(Show Context)
Citation Context ...hey are encountered in stochastic recursive utility optimization problems (see Antonelli [1], El Karoui et al. [10], Wang and Wu [23]) and principal-agent problems (see Williams [24], Cvitanic et al. =-=[9]-=-). Moreover, some financial optimization problems for large investors (see Cvitanic and Ma [8], Cucoo and Cvitanic [7], Buckdahn and Hu [3]) and some asset pricing problems with forward-backward diffe... |

5 |
Linear-quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system,”
- Yu
- 2012
(Show Context)
Citation Context ...(0), q(T ) = −cp(T ) + MTx(T ), y(T ) = cx(T ), k(T ) = 0, x(t) = ξ(t), p(t) = 0, t ∈ [−δ, 0); q(t) = 0, k(t) = 0, y(t) = ϕ(t), t ∈ (T, T + δ]. (4.6) This is an AFBSDDE with double dimensions. See Yu =-=[28]-=- for general theory of this kind equations without delay. We declare that if AFBSDDE (4.6) admits a unique solution (x(·), y(·), z(·), p(·), q(·), k(·)) then we can check that all the conditions in Th... |

5 |
The wellposedness of FBSDEs. Discrete Contin
- Zhang
(Show Context)
Citation Context ...ointed out, our results depend heavily on the G-monotonic assumption (H2). In fact, in the past several years, the research on the wellposedness of FBSDEs have made rapid progress. For example, Zhang =-=[29]-=- impose some kinds of simple weak-coupled conditions to obtain the existence and uniqueness for fully OPTIMAL CONTROL OF FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL DELAYED EQUATIONS 1095 coupled FBSDEs.... |

3 |
A type of generalized forward-backward stochastic differential equations and applications
- Chen, Wu
(Show Context)
Citation Context ...at the forward SDDE and backward ABSDE are fully coupled. The existence and uniqueness of the solution of such AFBSDDE under the above G-monotonic assumptions has been studied recently by Chen and Wu =-=[6]-=- (noting that this kind G-monotonic assumption was initially introduced by Hu and Peng [11], Peng and Wu [19]). Lemma 1.1 ([6]). Let (H1) and (H2)( or (H2)’) hold. Then for any v(·) ∈ Uad, AFBSDDE (1.... |

3 |
A maximum principle for optimal control problem of fully coupled forward-backward stochastic systems with partial information
- Meng
- 2009
(Show Context)
Citation Context ... dealing with time delay in Chen and Wu [5] and by Clarke generalized gradient’s approach dealing with sufficient conditions of optimality in Zhou [30]. We refer to Wu [25], Shi and Wu [21, 22], Meng =-=[13]-=-, Yong [26] for more details on maximum principles for fully coupled forward-backward stochastic systems without delay. The rest of this paper is organized as follows. In Section 2, we give the main r... |

3 |
The maximum principle for partially observed optimal control of fully coupled forward-backward stochastic system
- Shi, Wu
(Show Context)
Citation Context ...21], by methods dealing with time delay in Chen and Wu [5] and by Clarke generalized gradient’s approach dealing with sufficient conditions of optimality in Zhou [30]. We refer to Wu [25], Shi and Wu =-=[21, 22]-=-, Meng [13], Yong [26] for more details on maximum principles for fully coupled forward-backward stochastic systems without delay. The rest of this paper is organized as follows. In Section 2, we give... |