Some linear-quadratic stochastic differential games for equations in  Hilbert spaces with fractional Brownian motions

Tyrone E.  Duncan

doi:10.3934/dcds.2015.35.5435

Article Contents

2015, Volume 35, Issue 11: 5435-5445. Doi: 10.3934/dcds.2015.35.5435

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Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions

Tyrone E. Duncan^1,

1.
Department of Mathematics, University of Kansas, Lawrence, KS 66045

Received: November 30, 2012

Revised: January 31, 2014

Published: October 2015

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Abstract

A noncooperative, two person, zero sum, stochastic differential game is formulated and solved that is described by a linear stochastic equation in a Hilbert space with a fractional Brownian motion and a quadratic payoff functional for the two players. The stochastic equation can model stochastic partial differential equations not only with distributed strategies and noise but also with control strategies and noise restricted to the boundary of the domain. The optimal strategies for the two players are given explicitly. The verification method is a generalization of completion of squares and provides the optimal strategies directly without solving partial differential equations or backward stochastic differential equations. Some examples of games described by stochastic partial differential equations are given.

Keywords:

Stochastic differential games,
fractional Brownian motions,
stochastic partial differential equations.

Mathematics Subject Classification: 49N70, 91A25, 60G22, 60H15.

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