Equilibrium Strategies for Time-Inconsistent Stochastic Switching Systems

Mei, Hongwei; Yong, Jiongmin

doi:10.48550/arxiv.1712.09505

Cited by 5 publications

(4 citation statements)

References 13 publications

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“…It is worthy of pointing out that, to our best knowledge, representation PDEs of form (1.21) appeared the first time in the study of time-inconsistent optimal control problems ( [41], see also [35,25]). This indicates that there should be some intrinsic relationship between BSVIEs and time-inconsistent optimal control problems.…”

Section: The First Representation Pdementioning

confidence: 99%

Backward Stochastic Volterra Integral Equations--- Representation of Adapted Solutions

Wang¹,

Yong²

2018

Preprint

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For backward stochastic Volterra integral equations (BSVIEs, for short), under some mild conditions, the so-called adapted solutions or adapted M-solutions uniquely exist. However, satisfactory regularity of the solutions is difficult to obtain in general. Inspired by the decoupling idea of forward-backward stochastic differential equations, in this paper, for a class of BSVIEs, a representation of adapted M-solutions is established by means of the so-called representation partial differential equations and (forward) stochastic differential equations. Well-posedness of the representation partial differential equations are also proved in certain sense.

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Section: The First Representation Pdementioning

confidence: 99%

Backward Stochastic Volterra Integral Equations--- Representation of Adapted Solutions

Wang¹,

Yong²

2018

Preprint

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“…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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“…After the breakthrough in [24] and [12,11], there are a lot works on time-inconsistent control concerning MDPs and continuous-time models in the last decade (e.g. see [16,21,23,24,22,6,18]). Using a similar idea from [24], we will try to find the time-inconsistent equilibrium strategy.…”

Section: Introductionmentioning

confidence: 99%

Time-inconsistent Risk-sensitive Equilibrium for Countable-stated Markov Decision Processes

Mei¹

2019

Preprint

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This paper is devoted to solving a time-inconsistent risk-sensitive control problem with parameter ε and its limit case (ε → 0 + ) for countable-stated Markov decision processes (MDPs for short). Since the cost functional is time-inconsistent, it is impossible to find a global optimal strategy for both cases. Instead, for each case, we will prove the existence of time-inconstant equilibrium strategies which verify the so-called step-optimality. Moreover, we prove the convergence of ε-equilibriums and the corresponding value functions as ε → 0 + .

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Equilibrium Strategies for Time-Inconsistent Stochastic Switching Systems

Cited by 5 publications

References 13 publications

Backward Stochastic Volterra Integral Equations--- Representation of Adapted Solutions

Backward Stochastic Volterra Integral Equations--- Representation of Adapted Solutions

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

Time-inconsistent Risk-sensitive Equilibrium for Countable-stated Markov Decision Processes

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