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Two-player zero-sum stochastic differential games with random horizon

Part of: Miscellaneous topics in calculus of variations and optimal control Hamilton-Jacobi theories, including dynamic programming Game theory

Published online by Cambridge University Press:  15 November 2019

M. Ferreira ,
D. Pinheiro  and
S. Pinheiro
M. Ferreira*
Affiliation:
Universidade de Lisboa and Escola Superior de Hotelaria e Turismo, Instituto Politécnico do Porto
D. Pinheiro*
Affiliation:
Brooklyn College and The Graduate Center, City University of New York
S. Pinheiro*
Affiliation:
Queensborough Community College, City University of New York
*
*Postal address: CEMAPRE, ISEG, Universidade de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal. Email address: miguelferreira@esht.ipp.pt
**Postal address: Department of Mathematics, Brooklyn College, City University of New York, 2900 Bedford Avenue, Brooklyn, NY 11210, USA. Email address: dpinheiro@brooklyn.cuny.edu
***Postal address: Department of Mathematics and Computer Science, Queensborough Community College, City University of New York, 222-05, 56th Avenue, Bayside, NY 11364, USA. Email address: scoutopinheiro@qcc.cuny.edu
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Abstract

We consider a two-player zero-sum stochastic differential game with a random planning horizon and diffusive state variable dynamics. The random planning horizon is a function of a non-negative continuous random variable, which is assumed to be independent of the Brownian motion driving the state variable dynamics. We study this game using a combination of dynamic programming and viscosity solution techniques. Under some mild assumptions, we prove that the value of the game exists and is the unique viscosity solution of a certain nonlinear partial differential equation of Hamilton–Jacobi–Bellman–Isaacs type.

Keywords

stochastic differential gamedynamic programmingviscosity solutions

MSC classification

Primary: 49N70: Differential games
Secondary: 91A15: Stochastic games 49L20: Dynamic programming method 49L25: Viscosity solutions
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 4 , December 2019 , pp. 1209 - 1235
Copyright
© Applied Probability Trust 2019 

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