English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This paper studies the well-posedness of a class of nonlocal parabolicpartial differential equations (PDEs), or equivalently equilibriumHamilton-Jacobi-Bellman equations, which has a strong tie with thecharacterization of the equilibrium strategies and the associated valuefunctions for time-inconsistent stochastic control problems. Specifically, weconsider nonlocality in both time and space, which allows for modelling of thestochastic control problems with initial-time-and-state dependent objectivefunctionals. We leverage the method of continuity to show the globalwell-posedness within our proposed Banach space with our established Schauderprior estimate for the linearized nonlocal PDE. Then, we adopt a linearizationmethod and Banach's fixed point arguments to show the local well-posedness ofthe nonlocal fully nonlinear case, while the global well-posedness isattainable provided that a very sharp a-priori estimate is available. On top ofthe well-posedness results, we also provide a probabilistic representation ofthe solutions to the nonlocal fully nonlinear PDEs and an estimate on thedifference between the value functions of sophisticated and na\"{i}vecontrollers. Finally, we give a financial example of time inconsistency that isproven to be globally solvable.

On the Well-posedness of Hamilton-Jacobi-Bellman Equations of the Equilibrium Type