Stochastic Linear Quadratic Optimal Control Problems

Chen, S.; Yong, Jiongmin

doi:10.1007/s002450010016

Cited by 88 publications

(52 citation statements)

References 18 publications

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“…More generally, N can also be possibly negativethis is the so-called indefinite case. On these features, the interested reader is referred to Chen and Yong [3], Hu and Zhou [6], Kohlmann and Tang [7,10], Yong and Zhou [21], and the references therein.…”

Section: Preliminariesmentioning

confidence: 99%

Dynamic Programming for General Linear Quadratic Optimal Stochastic Control with Random Coefficients

Tang¹

2015

SIAM J. Control Optim.

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Abstract. We are concerned with the linear-quadratic optimal stochastic control problem where all the coefficients of the control system and the running weighting matrices in the cost functional are allowed to be predictable (but essentially bounded) processes and the terminal state-weighting matrix in the cost functional is allowed to be random. Under suitable conditions, we prove that the value field V (t, x, ω), (t, x, ω) ∈ [0, T ] × R n × Ω, is quadratic in x, and has the following form: V (t, x) = Ktx, x where K is an essentially bounded nonnegative symmetric matrix-valued adapted processes. Using the dynamic programming principle (DPP), we prove that K is a continuous semimartingale of the formwith k being a continuous process of bounded variation and (2003), pp. 53-75] via the stochastic maximum principle with a viewpoint of stochastic flow for the associated stochastic Hamiltonian system. The present paper is its companion, and gives the second but more comprehensive (seemingly much simpler, but appealing to the advanced tool of Doob-Meyer decomposition theorem, in addition to the DDP) adapted solution to a general BSRE via the DDP. Further extensions to the jump-diffusion control system and to the general nonlinear control system are possible.Key words. linear quadratic optimal stochastic control, random coefficients, Riccati equation, backward stochastic differential equations, dynamic programming, semi-martingale AMS subject classifications. 93E20, 49K45, 49N10, 60H101. Formulation of the problem and basic assumptions. Consider the following linear quadratic optimal stochastic control (SLQ in short form) problem: minimize over u ∈ L

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Section: Preliminariesmentioning

confidence: 99%

Dynamic Programming for General Linear Quadratic Optimal Stochastic Control with Random Coefficients

Tang¹

2015

SIAM J. Control Optim.

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“…Further, even if n = m, P (·) and (·) are not necessarily symmetric matrix valued. Thus, (4.9) is quite different from those derived from stochastic linear quadratic optimal control [2,[4][5][6][7]13,16,21,24]. We also note that Riccati BSDE (4.9) contains singularities in P (·) and has quadratic growth in (·).…”

Section: Proposition 41 Suppose the Following Bsde For Lr M×n -Valumentioning

confidence: 93%

Linear forward-backward stochastic differential equations with random coefficients

Yong

2005

Probab. Theory Relat. Fields

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Solvability of linear forward-backward stochastic differential equations (FBSDEs, for short) with random coefficients is studied. A decoupling reduction method is introduced via which a large class of linear FBSDEs with random or deterministic timevarying coefficients is proved to be solvable. On the other hand, by means of Four Step Scheme, a Riccati backward stochastic equation (BSDE, for short) for (m×n) matrix-valued processes is derived. Global solvability of such Riccati BSDEs is discussed for some special (but nontrivial) cases, which leads to the solvability of the corresponding linear FBSDEs.

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“…Problem (SLQ) has recently been extensively studied in [3,4], which is referred to as the stochastic linear quadratic optimal control problem (SLQ problem, for short) with random coefficient. Problem (SLQ) is said to be finite at …”

Section: Stochastic Lq Problem: a New Phenomenonmentioning

confidence: 99%