Stochastic recursive optimal control problem with time delay and applications

Shi, Jingtao; Jin, Xu; Zhang, Huanshui

doi:10.3934/mcrf.2015.5.859

Cited by 13 publications

(23 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In general, the optimal control problems are infinite-dimensional since the value function may depend on the initial path in a complicated way. The conditions under which optimal control problems with delay become finite-dimensional have been extensively discussed in the dynamic programming principle approach (see, for example, Elsanosi et al, 2000, Lassen and Risebro, 2003, David, 2008, Shi, 2013. Simply speaking, to make the problems finite-dimensional, it is required that the value function depends only on the initial path x 0 (·) through the following two functionals…”

Section: Discussion On the Solvability Of The Problemmentioning

confidence: 99%

“…Current research on this topic can be divided into two directions. One direction involves a system of three-coupled adjoint equations, which consists of two BSDEs and one backward ordinary differential equation (ODE), see for example, Øksendal and Sulem (2000), David (2008), Agram et al (2012) and Shi (2013). And another direction, the adjoint equation is given by a time-advanced BSDE.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance

2014

View full text Add to dashboard Cite

This paper investigates a stochastic optimal control problem with delay and of mean-field type, where the controlled state process is governed by a mean-field jump-diffusion stochastic delay differential equation. Two sufficient maximum principles and one necessary maximum principle are established for the underlying systems. As an application, a bicriteria mean-variance portfolio selection problem with delay is studied. Under certain conditions, explicit expressions are provided for the efficient portfolio and the efficient frontier, which are as elegant as those in the classical mean-variance problem without delays.

show abstract

Section: Discussion On the Solvability Of The Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance

2014

View full text Add to dashboard Cite

show abstract

“…Therefore, it is of great relevance to develop the maximum principles under control systems with delay and apply these maximum principles to solve practical problems arising in the real world. See, for example, [16,17,[19][20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

“…Meng, Y. Shen / Journal of Computational and Applied Mathematics 279 (2015)[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] …”

mentioning

confidence: 99%

Optimal control of mean-field jump-diffusion systems with delay: A stochastic maximum principle approach

Meng

Shen

2015

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

a b s t r a c tThis paper is concerned with an optimal control problem under mean-field jump-diffusion systems with delay. Firstly, some existence and uniqueness results are proved for a jumpdiffusion mean-field stochastic delay differential equation and a jump-diffusion meanfield advanced backward stochastic differential equation. Then necessary and sufficient maximum principles for control systems of mean-field type and with delay are established under certain conditions. A mean-field, delayed, linear-quadratic control problem is finally discussed using the obtained maximum principles.

show abstract

“…The justification will be given in Remark 2 below. Using some properties of the BSDE and analysis technique, we expand the extension of the dynamic programming principle of the recursive control problem in [16,22,21,50,56,61,63,68] to the singular controls case. And then, we show that, provided the problem is formulated within a Markovian framework, the value function is a unique viscosity solution of the problem for one kind of nonlinear H-J-B inequality, in a given class of bounded and continuous functions.…”

Section: Introductionmentioning

confidence: 99%

Singular optimal controls of stochastic recursive systems and Hamilton–Jacobi–Bellman inequality

Zhang

2019

Journal of Differential Equations

View full text Add to dashboard Cite

In this paper, we study the optimal singular controls for stochastic recursive systems, in which the control has two components: the regular control, and the singular control. Under certain assumptions, we establish the dynamic programming principle for this kind of optimal singular controls problem, and prove that the value function is a unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman inequality, in a given class of bounded and continuous functions. At last, an example is given for illustration.T ] → R n×m are given deterministic functions, (W s ) s≥0 is an d-dimensional Brownian motion, (x, t) are initial time and state, v (·) : [0, T ] → R k is classical control process, and ξ (·) : [0, T ] → R m , with nondecreasing left-continuous with right limits stands for the singular control.The aim is to minimize the cost functional:where. . . m} are given deterministic functions, where l (·) represents the running cost tare of the problem and K (·) the cost rate of applying the singular control. [40], Karatzas and Shreve [43], and Menaldi and Taksar [52]. The second one is related to probabilistic methods; see Baldursson [6], Boetius [8, 9], Boetius and Kohlmann [10], El Karoui and Karatzas [29, 30], Karatzas [39], and Karatzas and Shreve [41, 42]. Third, the DPP, has been studied in a general context, for example, by Boetius [9], Haussmann and Suo [36], Fleming and Soner [31], and Zhu [66]. At last the maximum principle for optimal singular controls (see, for example, Cadenillas and Haussmann [17], Dufour and Miller [26], Dahl and Øksendal [27] see references therein). The existence for optimal singular control can be found in Haussmann and Suo [36] and Dufour and Miller [25] via different approaches. It is necessary to point out that singular control problems are largely used in diverse areas such as mathematical finance (see Baldursson and Karatzas [12], Chiarolla and Haussmann [18], Kobila [44], and Karatzas and Wang [45], Davis and Norman [24]), manufacturing systems (see, Shreve, Lehoczky, and Gaver [60]), and queuing systems (see Martins and Kushner [53]). Particularly, the application of H-J-B inequality in finance can be seen in Pagès and Possamaï [58], which is employed to investigate the bank monitoring incentives.As is well known for the classical stochastic control problems, DPP is satisfied. Moreover, if the value function has appropriate regularity, it admits a second-order nonlinear partial differential equation (H-J-B equation) (see Fleming and Rishel [32] and Lions [49] reference therein). In the frame work of singular stochastic control, the H-J-B equation becomes a second-order variational inequality (see Fleming and Soner [31], Haussmannand, Suo [35]).Wang [63] firstly introduces and studies a class of singular control problems with recursive utility, where the cost function is determined by a backward stochastic differential equation (BSDE in short). More preciously, the cost functional is defined bywhere Y t,x;v,ξ t is determined by    dX t,x;ξ s = aX t,x;ξ s + b ds + ...

show abstract

Stochastic recursive optimal control problem with time delay and applications

Cited by 13 publications

References 43 publications

Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance

Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance

Optimal control of mean-field jump-diffusion systems with delay: A stochastic maximum principle approach

Singular optimal controls of stochastic recursive systems and Hamilton–Jacobi–Bellman inequality

Contact Info

Product

Resources

About