Probabilistic interpretation of a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations

By introducing a stochastic differential game whose dynamics and multi-dimensional cost functionals form a multi-dimensional coupled forward-backward stochastic differential equation with jumps, we give a probabilistic interpretation to a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations. F...

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Veröffentlicht in:	ESAIM. Control, optimisation and calculus of variations optimisation and calculus of variations, 2021, Vol.27, p.S17
Hauptverfasser:	Li, Juan, Li, Wenqiang, Wei, Qingmeng
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Sprache:	eng
Schlagworte:	Differential equations Differential games Dynamic programming Mathematical analysis Probability theory Random variables
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container_title	ESAIM. Control, optimisation and calculus of variations
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creator	Li, Juan Li, Wenqiang Wei, Qingmeng
description	By introducing a stochastic differential game whose dynamics and multi-dimensional cost functionals form a multi-dimensional coupled forward-backward stochastic differential equation with jumps, we give a probabilistic interpretation to a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations. For this, we generalize the definition of the lower value function initially defined only for deterministic times t and states x to stopping times τ and random variables η ∈ L 2 (Ω, τ, P ; ℝ). The generalization plays a key role in the proof of a strong dynamic programming principle. This strong dynamic programming principle allows us to show that the lower value function is a viscosity solution of our system of multi-dimensional coupled Hamilton-Jacobi-Bellman-Isaacs equations. The uniqueness is obtained for a particular but important case.
doi_str_mv	10.1051/cocv/2020070
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subjects	Differential equations Differential games Dynamic programming Mathematical analysis Probability theory Random variables
title	Probabilistic interpretation of a system of coupled Hamilton-Jacobi-Bellman-Isaacs equations
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