Parabolic Bellman-Systems with Mean Field Dependence

Alain Bensoussan; Dominic Breit; Jens Frehse

doi:10.1007/s00245-016-9344-6

Parabolic Bellman-Systems with Mean Field Dependence

Alain Bensoussan, Dominic Breit^*, Jens Frehse

^*Corresponding author for this work

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

4 Citations (Scopus)

Abstract

We consider the necessary conditions for Nash-points of Vlasov-McKean functionals Ji[v]=∫Qmfi(·,m,v)dxdt (i= 1 ,.. , N). The corresponding payoffs fⁱ depend on the controls v and, in addition, on the field variable m= m(v). The necessary conditions lead to a coupled forward-backward system of nonlinear parabolic equations, motivated by stochastic differential games. The payoffs may have a critical nonlinearity of quadratic growth and any polynomial growth w.r.t. m is allowed as long as it can be dominated by the controls in a certain sense. We show existence and regularity of solutions to these mean-field-dependent Bellman systems by a purely analytical approach, no tools from stochastics are needed.

Original language	English
Pages (from-to)	419-432
Journal	Applied Mathematics and Optimization
Volume	73
Issue number	3
DOIs	https://doi.org/10.1007/s00245-016-9344-6
Publication status	Published - 1 Jun 2016

Research Keywords

Bellman equations
Mean field dependence
Nonlinear parabolic systems
Stochastic differential games

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title = "Parabolic Bellman-Systems with Mean Field Dependence",

abstract = "We consider the necessary conditions for Nash-points of Vlasov-McKean functionals Ji[v]=∫Qmfi(·,m,v)dxdt (i= 1 ,.. , N). The corresponding payoffs fi depend on the controls v and, in addition, on the field variable m= m(v). The necessary conditions lead to a coupled forward-backward system of nonlinear parabolic equations, motivated by stochastic differential games. The payoffs may have a critical nonlinearity of quadratic growth and any polynomial growth w.r.t. m is allowed as long as it can be dominated by the controls in a certain sense. We show existence and regularity of solutions to these mean-field-dependent Bellman systems by a purely analytical approach, no tools from stochastics are needed.",

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T1 - Parabolic Bellman-Systems with Mean Field Dependence

AU - Bensoussan, Alain

AU - Breit, Dominic

AU - Frehse, Jens

PY - 2016/6/1

Y1 - 2016/6/1

N2 - We consider the necessary conditions for Nash-points of Vlasov-McKean functionals Ji[v]=∫Qmfi(·,m,v)dxdt (i= 1 ,.. , N). The corresponding payoffs fi depend on the controls v and, in addition, on the field variable m= m(v). The necessary conditions lead to a coupled forward-backward system of nonlinear parabolic equations, motivated by stochastic differential games. The payoffs may have a critical nonlinearity of quadratic growth and any polynomial growth w.r.t. m is allowed as long as it can be dominated by the controls in a certain sense. We show existence and regularity of solutions to these mean-field-dependent Bellman systems by a purely analytical approach, no tools from stochastics are needed.

AB - We consider the necessary conditions for Nash-points of Vlasov-McKean functionals Ji[v]=∫Qmfi(·,m,v)dxdt (i= 1 ,.. , N). The corresponding payoffs fi depend on the controls v and, in addition, on the field variable m= m(v). The necessary conditions lead to a coupled forward-backward system of nonlinear parabolic equations, motivated by stochastic differential games. The payoffs may have a critical nonlinearity of quadratic growth and any polynomial growth w.r.t. m is allowed as long as it can be dominated by the controls in a certain sense. We show existence and regularity of solutions to these mean-field-dependent Bellman systems by a purely analytical approach, no tools from stochastics are needed.

KW - Bellman equations

KW - Mean field dependence

KW - Nonlinear parabolic systems

KW - Stochastic differential games

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UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84964553145&origin=recordpage

U2 - 10.1007/s00245-016-9344-6

DO - 10.1007/s00245-016-9344-6

M3 - RGC 21 - Publication in refereed journal

SN - 0095-4616

VL - 73

SP - 419

EP - 432

JO - Applied Mathematics and Optimization

JF - Applied Mathematics and Optimization

IS - 3

ER -

Parabolic Bellman-Systems with Mean Field Dependence

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