Equilibrium strategies for time-inconsistent stochastic switching systems

Mei, Hongwei; Yong, Jiongmin

doi:10.1051/cocv/2018051

Cited by 26 publications

(24 citation statements)

References 47 publications

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“…The idea can be at least traced back to the work of Pollak [31] in 1968. Later, the approach was adopted and further developed by Ekeland-Lazrak [14,15]; Yong [45][46][47]; Björk-Murgoci [6]; Björk-Murgoci-Zhou [7]; Björk-Khapko-Murgoci [5]; Wei-Yong-Yu [42], Mei-Yong [28], and Yan-Yong [43] for various kinds of problems.…”

Section: Bsviesmentioning

confidence: 99%

Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations

Wang

Yong

2021

ESAIM: COCV

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An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situation with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate equilibrium strategies is constructed associated with partitions of the time intervals. By sending the mesh size of the time interval partition to zero, an equilibrium Hamilton--Jacobi--Bellman (HJB, for short) equation is derived, through which the equilibrium valued function and an equilibrium strategy are obtained. Under certain conditions, a verification theorem is proved and the well-posedness of the equilibrium HJB is established. As a sort of Feynman-Kac formula for the equilibrium HJB equation, a new class of BSVIEs (containing the diagonal values $Z(r,r)$ of $Z(\cd\,,\cd)$) is naturally introduced and the well-posedness of such kind of equations is briefly discussed.

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Section: Bsviesmentioning

confidence: 99%

Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations

Wang

Yong

2021

ESAIM: COCV

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“…By (4.15)-(4.16), (Ỹ t,x (·, ·),Z t,x (·, ·)) also satisfies the EBSVIE (1.17) and is also F-adapted. By the uniqueness of the adapted solutions to EBSVIE (1.17), we have 17) which means that for any s…”

Section: Regularity Of the Adapted Solutionmentioning

confidence: 99%

“…To our best knowledge, the PDEs of form (1.8) appeared the first time in the study of time-inconsistent optimal control problems. In the time-inconsistent optimal control problems, the PDE (1.8) serves as an equilibrium HJB equation, which is used to express the equilibrium strategy and equilibrium vale function ( [37], see also [34], [17]).…”

Section: Introductionmentioning

confidence: 99%

Extended backward stochastic Volterra integral equations, Quasilinear parabolic equations, and Feynman–Kac formula

Wang

2020

Stoch. Dyn.

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In this paper, we establish the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. We first introduce the extended backward stochastic Volterra integral equations (EBSVIEs, for short). Under some mild conditions, we establish the well-posedness of EBSVIEs and obtain some regularity results of the adapted solution to the EBSVIEs via Malliavin calculus. We show that a given function expressed in terms of the solution to the EBSVIEs solves a certain system of non-local parabolic partial differential equations (PDEs, for short), which generalizes the famous nonlinear Feynman-Kac formula in Pardoux-Peng [21].Let us look at some special cases of EBSVIE (1.1). Suppose g(t, s, y, y ′ , z) = g(t, s, y, z), ∀(t, s, y, y ′ , z)then EBSVIE (1.1) is reduced to the following form:which is a family of so-called backward stochastic differential equations (BSDEs, for short) parameterized by t ∈ [0, T ]; see [20,12,16,39] for systematic discussions of BSDEs.On the other hand, if, then EBSVIE (1.1) is reduced to the following form:which is a so-called backward stochastic Volterra integral equation (BSVIE, for short). This is exactly why we call (1.1) an extended backward stochastic Volterra integral equation. BSVIEs of the form (1.3) was initially studied by Lin [15] and followed by several other researchers: Aman and NZi [3], Yong [35], Ren [24], Anh, Grecksch, and Yong [4], Djordjevi'c and Jankovi'c [6, 7], Hu and Øksendal [10], and the references therein. Recently, Wang, Sun, and Yong [28] established the well-posedness of quadratic BSVIEs (which means the generator g(t, s, y, z) of (1.3) has a quadratic growth in z) and explored the applications of quadratic BSVIEs to equilibrium dynamic risk measure and equilibrium recursive utility process. BSVIE of the more general form Y (t) = ψ(t) + T t g(t, r, Y (r), Z(t, r), Z(r, t))dr − T t Z(t, r)dW (r) (1.4)was firstly introduced by Yong [36] in his research on optimal control of forward stochastic Volterra integral equations (FSVIEs, for short). The BSVIE (1.4) has a remarkable feature that its solution might not be unique due to lack of restriction on the term Z(r, t); 0 ≤ t ≤ r ≤ T . Suggested by the nature of the equation from the adjoint equation in the Pontryagin type maximum principle, Yong [36] introduced the notion of adapted M-solution: A pair (Y (·), Z(· , ·)) is called an adapted M-solution to (1.4), if in addition to (i)-(iii) stated above, the following condition is also satisfied:a.e. t ∈ [0, T ], a.s. (1.5) Under usual Lipschitz conditions, well-posedness was established in [36] for the adapted M-solutions to BSVIEs of form (1.4). This important development has triggered extensive research on BSVIEs and their applications. For instance, Anh, Grecksch and Yong [4] investigated BSVIEs in Hilbert spaces; Shi, Wang and Yong [25] studied well-posedness of BSVIEs containing mean-fields (of the unknowns); Ren [24], Wang and Zhang [33] discussed BSVIEs with ...

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“…Hence, by the uniqueness of the solution to (32), we obtain that p t (s) , q t (s) = p t (s) ,q t (s) , a.s., for all s ∈ [t, T ] .…”

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confidence: 94%

“…Yong [43] performed a multi-person differential game approach for a general discounting time-inconsistent stochastic optimal control problem and he characterized the Nash equilibrium controls via the so-called equilibrium HJB equation. Further work devoted to Yong's approach can be found in [44], [38], [41], [40] and [32]. In a series of papers, Basak and Chabakauri [7], Zeng et al [46], Böjrk et al [9] derived feedback Nash equilibrium solutions to various dynamic optimization problems under the mean-variance criterion by solving an extended HJB system.…”

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confidence: 99%

Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach

Alia¹

2020

Mathematical Control &Amp; Related Fields

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In this paper, we investigate a class of time-inconsistent stochastic control problems for stochastic differential equations with deterministic coefficients. We study these problems within the game theoretic framework, and look for open-loop Nash equilibrium controls. Under suitable conditions, we derive a verification theorem for equilibrium controls via a flow of forwardbackward stochastic partial differential equations. To illustrate our results, we discuss a mean-variance problem with a state-dependent trade-off between the mean and the variance.2010 Mathematics Subject Classification. Primary: 91B51, 93E20, 60H30, 93E99, 60H10; Secondary: 93E25.

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Equilibrium strategies for time-inconsistent stochastic switching systems

Cited by 26 publications

References 47 publications

Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations

Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations

Extended backward stochastic Volterra integral equations, Quasilinear parabolic equations, and Feynman–Kac formula

Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach

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