Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T15:11:32.477Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bayraktar, E. and Sîrbu, M. (2012). Stochastic Perron’s method and verification without smoothness using viscosity comparison: the linear case. Proc. Amer. Math. Soc. 140, 3645–3654.CrossRefGoogle Scholar
Bayraktar, E. and Sîrbu, M. (2013). Stochastic Perron’s method for Hamilton–Jacobi–Bellman equations. SIAM J. Control Optim. 51, 4274–4294.CrossRefGoogle Scholar
Bayraktar, E. and Sîrbu, M. (2014). Stochastic Perron’s method and verification without smoothness using viscosity comparison: obstacle problems and Dynkin games. Proc. Amer. Math. Soc. 142, 1399–1412.CrossRefGoogle Scholar
Bayraktar, E. and Yao, S. (2013). A weak dynamic programming principle for zero-sum stochastic differential games with unbounded controls. SIAM J. Control Optim. 51, 2036–2080.CrossRefGoogle Scholar
Berkovitz, L. and Fleming, W. (1955). On Differential Games with Integral Payoff (Ann. Math. Study 39). Princeton University Press, Princeton, NJ.Google Scholar
Biswas, I. (2012). On zero-sum stochastic differential games with jump-diffusion driven state: a viscosity solution framework. SIAM J. Control Optim. 50, 1823–1858.CrossRefGoogle Scholar
Blanchet-Scalliet, C., El Karoui, N., Jeanblanc, M. and Martellini, L. (2008). Optimal investment decisions when time-horizon is uncertain. J. Math. Econom. 44, 1100–1113.CrossRefGoogle Scholar
Bruhn, K. and Steffensen, M. (2011). Household consumption, investment and life insurance. Insurance Math. Econom. 48, 315–325.CrossRefGoogle Scholar
Buckdahn, R. and Li, J. (2008). Stochastic differential games and viscosity solutions of Hamilton–Jacobi–Bellman–Isaacs equations. SIAM J. Control Optim. 47, 444–475.CrossRefGoogle Scholar
Buckdahn, R., Li, J. and Quincampoix, M. (2014). Value in mixed strategies for zero-sum stochastic differential games without Isaacs condition. Ann. Prob. 42, 1724–1768.CrossRefGoogle Scholar
Cardaliaguet, P. and Rainer, C. (2009). Stochastic differential games with asymmetric information. Appl. Math. Optim. 59, 1–36.CrossRefGoogle Scholar
Crandall, M. and Lions, P.-L. (1983). Viscosity solutions of Hamilton–Jacobi equations. Trans. Amer. Math. Soc. 277, 1–42.CrossRefGoogle Scholar
Crandall, M., Ishii, H. and Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27, 1–67.CrossRefGoogle Scholar
Duarte, I., Pinheiro, D., Pinto, A. and Pliska, S. (2014). Optimal life insurance purchase, consumption and investment on a financial market with multi-dimensional diffusive terms. Optimization 63, 1737–1760.CrossRefGoogle Scholar
Elliott, R. and Kalton, N. (1972). The Existence of Value in Differential Games (Mem. Amer. Math. Soc. 126). American Mathematical Society.Google Scholar
Evans, L. and Souganidis, P. (1984). Differential games and representation formulas for solutions of Hamilton–Jacobi–Isaacs equations. Indiana Univ. Math. J. 33, 773–797.CrossRefGoogle Scholar
Fleming, W. and Souganidis, P. (1989). On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38, 293–314.CrossRefGoogle Scholar
Friedman, A. (1971). Differential Games. Wiley, New York, NY.Google Scholar
Friedman, A. (1972). Stochastic differential games. J. Differential Equations 11, 79–108.CrossRefGoogle Scholar
Hamadène, S. and Mu, R. (2015). Existence of Nash equilibrium points for Markovian non-zero-sum stochastic differential games with unbounded coefficients. Stochastics 87, 85–111.CrossRefGoogle Scholar
Isaacs, R. (1965). Differential Games: A Mathematical Theory with Applications to Warfare. Wiley, New York, NY.Google Scholar
Ishii, H. (1989). On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Comm. Pure Appl. Math. 42, 15–45.CrossRefGoogle Scholar
Ishii, H. and Lions, P.-L. (1990). Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 83, 26–78.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. (2000). Brownian Motion and Stochastic Calculus. Springer, New York, Heidelberg and Berlin.Google Scholar
Katsoulakis, M. (1995). A representation formula and regularizing properties for viscosity solutions of second-order fully nonlinear degenerate parabolic equations. Nonlinear Anal. 24, 147–158.CrossRefGoogle Scholar
Kumar, K. (2008). Nonzero sum stochastic differential games with discounted payoff criterion: an approximating Markov chain approach. SIAM J. Control Optim. 47, 374–395.CrossRefGoogle Scholar
Kwak, M., Yong, H. and Choi, U. (2009). Optimal investment and consumption decision of family with life insurance. Insurance Math. Econom. 48, 176–188.CrossRefGoogle Scholar
Mao, X. (2007). Stochastic Differential Equations and Applications, 2nd edn. Horwood Publishing, Chichester, UK.Google Scholar
Marín-Solano, J. and Shevkoplyas, E. (2011). Non-constant discounting and differential games with random time horizon. Automatica 47, 2626–2638.CrossRefGoogle Scholar
Mousa, A., Pinheiro, D. and Pinto, A. (2016). Optimal life insurance purchase from a market of several competing life insurance providers. Insurance Math. Econom. 67, 133–144.CrossRefGoogle Scholar
Nisio, M. (1988). Stochastic differential games and viscosity solutions of Isaacs equations. Nagoya Math. J. 110, 163–184.CrossRefGoogle Scholar
Petrosyan, L. and Murzov, N. (1966). Game-theoretic problems of mechanics. Litovsk. Math. Sb. 6, 423–433.Google Scholar
Petrosyan, L. and Shevkoplyas, E. (2000). Cooperative games with random duration. Vestnik of St. Petersburg Univ. Series 1 4, 18–23.Google Scholar
Petrosyan, L. and Shevkoplyas, E. (2003). Cooperative solutions for games with random duration. Game Theory and Applications IX, 125–139.Google Scholar
Pham, T. and Zhang, J. (2014). Two person zero-sum game in weak formulation and path dependent Bellman–Isaacs equation. SIAM J. Control Optim. 52, 2090–2121.CrossRefGoogle Scholar
Pliska, S. and Ye, J. (2007). Optimal life insurance purchase and consumption/investment under uncertain lifetime. J. Bank Finance 31, 1307–1319.CrossRefGoogle Scholar
Shen, Y. and Wei, J. (2016). Optimal investment-consumption-insurance with random parameters. Scand. Actuar. J. 2016, 37–62.CrossRefGoogle Scholar
Sîrbu, M. (2014). Stochastic Perron’s method and elementary strategies for zero-sum differential games. SIAM J. Control Optim. 52, 1693–1711.CrossRefGoogle Scholar
Souganidis, P. (1985). Approximation schemes for viscosity solutions of Hamilton–Jacobi equations. J. Differential Equations 59, 1–43.CrossRefGoogle Scholar
Souganidis, P. (1985). Max-min representations and product formulas for the viscosity solutions of Hamilton–Jacobi equations with applications to differential games. Nonlinear Anal. 9, 217–257.CrossRefGoogle Scholar
Tang, S. and Hou, S. (2007). Switching games of stochastic differential systems. SIAM J. Control Optim. 46, 900–929.CrossRefGoogle Scholar
Yaari, M. (1965). Uncertain lifetime, life insurance and the theory of the consumer. Rev. Econom. Stud. 32, 137–150.CrossRefGoogle Scholar