Singular Stochastic Control, Linear Diffusions, and Optimal Stopping: A Class of Solvable Problems

Alvarez, Luis H. R.

doi:10.1137/s0363012900367825

Cited by 88 publications

(84 citation statements)

References 19 publications

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“…Mathematical analysis of such control problems has evolved considerably from the initial heuristics to the more sophisticated and standard stochastic control approach, and from the very special case study to general payoff functions. (See Harrison and Taksar (1983); Karatzas (1985); Karatzas and Shreve (1985); El Karoui and Karatzas ( , 1989; Ma (1992); Zervos (1994, 1998); Boetius and Kohlmann (1998); Alvarez (2000Alvarez ( , 2001; Bank (2005); Boetius (2005)). Most recently, Merhi and Zervos (2007) analyzed this problem in great generality and provided explicit solutions for the special case where the payoff is of Cobb-Douglas type.…”

Section: Introductionmentioning

confidence: 99%

A Class of Singular Control Problems and the Smooth Fit Principle

Guo¹,

Tomecek

2009

SIAM J. Control Optim.

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This paper analyzes a class of singular control problems for which value functions are not necessarily smooth. Necessary and sufficient conditions for the well-known smooth fit principle, along with the regularity of the value functions, are given. Explicit solutions for the optimal policy and for the value functions are provided. In particular, when payoff functions satisfy the usual Inada conditions, the boundaries between action and no-action regions are smooth and strictly monotonic as postulated and exploited in the existing literature (Dixit and Pindyck (1994)

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Section: Introductionmentioning

confidence: 99%

A Class of Singular Control Problems and the Smooth Fit Principle

Guo¹,

Tomecek

2009

SIAM J. Control Optim.

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“…The stochastic singular control problems have received considerable research attention in recent years due to wide applicability in a number of different areas, see for instance [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] . In most classical cases, the optimal singular control problem was investigated through dynamic programming principle.…”

Section: L(t)dξ(t)mentioning

confidence: 99%

“…Stochastic maximum principle for optimal control problems of forward backward systems involving impulse controls has been studied in Wu and Zhang [3,12]. The stochastic maximum principle for singular control was considered by many authors, see for instance [1,2,[4][5][6][7][8][9][10]. The first version of maximum principle for singular stochastic control problems was obtained by Cadenillas and Haussmann [9].…”

Section: L(t)dξ(t)mentioning

confidence: 99%

Singular mean-field optimal control for forward-backward stochastic systems and applications to finance

Hafayed

2014

Int. J. Dynam. Control

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In this paper, we study a class of singular stochastic optimal control problems for systems described by mean-field forward-backward stochastic differential equations, in which the coefficient depend not only on the state process but also its marginal law of the state process through its expected value. Moreover, the cost functional is also of mean-field type. The control variable has two components, the first being absolutely continuous and the second singular control. Necessary conditions for optimal control for this systems in the form of a Pontrygin maximum principle are established by means convex perturbation techniques for both continuous and singular parts. Our stochastic maximum principle differs from the classical one in the sense that here the adjoint equation has a mean-field type. The control domain is assumed to be convex. As an illustration of our results, we consider a mean-variance portfolio selection mixed with a recursive utility functional optimization problem involving singular control. The explicit expression of the optimal portfolio selection strategy is obtained in the state feedback form involving both state process and its marginal distribution, via the solutions of Riccati ordinary differential equations with time-inconsistent solution.Keywords Singular stochastic control · Mean-field forward-backward stochastic differential equation · Mean-field type maximum principle · Time-inconsistent control problem · McKean-Vlasov systems M. Hafayed

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“…Lf (x, u) def = ∇f (x), m(x, u) + 1 2 tr σ(x, u)σ T (x, u)∇ 2 f (x) (2) for f ∈ C 2 (R d ). We may write L u f (x) for Lf (u, x), treating u as a parameter.…”

Section: Weak Solutionmentioning

confidence: 99%

“…This research originated in heavy traffic limits of controlled queues [60], [107]. See [2], [35], [51], [74], [75] for some recent contributions and applications to finance.…”

Section: Other Problemsmentioning

confidence: 99%

Controlled diffusion processes

Borkar¹

2005

Probab. Surveys

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This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.

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Singular Stochastic Control, Linear Diffusions, and Optimal Stopping: A Class of Solvable Problems

Cited by 88 publications

References 19 publications

A Class of Singular Control Problems and the Smooth Fit Principle

A Class of Singular Control Problems and the Smooth Fit Principle

Singular mean-field optimal control for forward-backward stochastic systems and applications to finance

Controlled diffusion processes

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