A regular equilibrium solves the extended HJB system

Lindensjö, Kristoffer

doi:10.1016/j.orl.2019.07.011

Cited by 21 publications

(20 citation statements)

References 35 publications

(101 reference statements)

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“…Recently, there has been a substantial effort to develop the literature on the gametheoretic approach to time-inconsistent control problems. The main theoretical result of time-inconsistent Markovian stochastic control is a characterization of an equilibrium as a solution to a generalized HJB equation called the extended HJB system, see [4,5,29]. Recently a considerable literature using the extended HJB system to study time-inconsistent control problems has emerged, examples include [2,6,17,26,28].…”

Section: Previous Literaturementioning

confidence: 99%

On time-inconsistent stopping problems and mixed strategy stopping times

Christensen

Lindensjö

2020

Stochastic Processes and their Applications

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A game-theoretic framework for time-inconsistent stopping problems where the time-inconsistency is due to the consideration of a non-linear function of an expected reward is developed. A class of mixed strategy stopping times that allows the agents in the game to jointly choose the intensity function of a Cox process is introduced and motivated. A subgame perfect Nash equilibrium is defined. The equilibrium is characterized and other necessary and sufficient equilibrium conditions including a smooth fit result are proved. Existence and uniqueness are investigated. A mean-variance and a variance problem are studied. The state process is a general one-dimensional Itô diffusion.

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Section: Previous Literaturementioning

confidence: 99%

On time-inconsistent stopping problems and mixed strategy stopping times

Christensen

Lindensjö

2020

Stochastic Processes and their Applications

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“…The main feature of that theory is a generalization of the standard HJB equation called the extended HJB system and the main result is a verification theorem saying that if a solution to the extended HJB system exists then it corresponds to an equilibrium. In [27] it is shown that a regular equilibrium is necessarily a solution to an extended HJB system. Other papers studying specific time-inconsistent control problems from a more mathematical perspective include [7,13,17,22,26].…”

Section: Previous Literaturementioning

confidence: 99%

On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems

Christensen¹,

Lindensjö²

2018

SIAM J. Control Optim.

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Standard Markovian optimal stopping problems are consistent in the sense that the first entrance time into the stopping set is optimal for each initial state of the process. Clearly, the usual concept of optimality cannot in a straightforward way be applied to non-standard stopping problems without this time-consistent structure. This paper is devoted to the solution of time-inconsistent stopping problems with the reward depending on the initial state using an adaptation of Strotz's consistent planning. More precisely, we give a precise equilibrium definition -of the type subgame perfect Nash equilibrium based on pure Markov strategies. In general, such equilibria do not always exist and if they exist they are in general not unique. We, however, develop an iterative approach to finding equilibrium stopping times for a general class of problems and apply this approach to one-sided stopping problems on the real line. We furthermore prove a verification theorem based on a set of variational inequalities which also allows us to find equilibria. In the case of a standard optimal stopping problem, we investigate the connection between the notion of an optimal and an equilibrium stopping time. As an application of the developed theory we study a selling strategy problem under exponential utility and endogenous habit formation.

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“…and K s,x,z (τ ) = ĝ τ, X s,x,z (τ ) , Xs,z (τ ) = ĝ1 τ, X s,x,z (τ ) , Xs,z (τ ) , ..., ĝm τ, X s,x,z (τ ) , Xs,z (τ ) . (27) Note that by virtue of Definition 3.4 and Assumptions (H1)-(H2), it is not difficult to verify that all needed integrability conditions to the above representations are satisfied. Thus, setting τ = s in ( 26)-( 27), we get…”

Section: Ol Equilibriums For Time-inconsistent Stochastic Control Problems 15mentioning

confidence: 99%

“…Inspired by the discrete case, they first derived in an heuristic way the so-called extended HJB equations (that is a system of three-coupled fully nonlinear parabolic PDEs), and then rigorously proved a verification theorem. Recently, Lindensjö [27] derived rigourously the extended HJB equations without using arguments from the discrete-time case. He and Jiang [20] and Hernández and Possamaï [21] generalized [5] by refining the definition of the closed-loop equilibrium concept.…”

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Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems

Alia¹

2023

MCRF

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<p style='text-indent:20px;'>This paper studies open-loop equilibriums for a general class of time-inconsistent stochastic control problems under jump-diffusion SDEs with deterministic coefficients. Inspired by the idea of Four-Step-Scheme for forward-backward stochastic differential equations with jumps (FBSDEJs, for short), we derive two systems of integro-partial differential equations (IPDEs, for short). Then, we rigorously prove a verification theorem which provides a sufficient condition for open-loop equilibrium strategies. As special cases, a mean-variance portfolio selection problem and a time-inconsistent problem under non-exponential discounting are discussed.</p>

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A regular equilibrium solves the extended HJB system

Cited by 21 publications

References 35 publications

On time-inconsistent stopping problems and mixed strategy stopping times

On time-inconsistent stopping problems and mixed strategy stopping times

On Finding Equilibrium Stopping Times for Time-Inconsistent Markovian Problems

Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems

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