Abstract
We consider a network of d companies (insurance companies, for example) operating under a treaty to diversify risk. Internal and external borrowing are allowed to avert ruin of any member of the network. The amount borrowed to prevent ruin is viewed upon as control. Repayment of these loans entails a control cost in addition to the usual costs. Each company tries to minimize its repayment liability. This leads to a d -person differential game with state space constraints. If the companies are also in possible competition a Nash equilibrium is sought. Otherwise a utopian equilibrium is more appropriate. The corresponding systems of HJB equations and boundary conditions are derived. In the case of Nash equilibrium, the Hamiltonian can be discontinuous; there are d interlinked control problems with state constraints; each value function is a constrained viscosity solution to the appropriate discontinuous HJB equation. Uniqueness does not hold in general in this case. In the case of utopian equilibrium, each value function turns out to be the unique constrained viscosity solution to the appropriate HJB equation. Connection with Skorokhod problem is briefly discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Atar, R., Budhiraja, A.: Singular control with state constraints on unbounded domain. Ann. Probab. 34, 1864–1909 (2006)
Atar, R., Dupuis, P., Shwartz, A.: An escape-time criterion for queueing networks: asymptotic risk-sensitive control via differential games. Math. Oper. Res. 28, 801–835 (2003)
Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhauser, Boston (1997)
Bensoussan, A., Frehse, J.: Stochastic games for N players. J. Optim. Theory Appl. 105, 543–565 (2000)
Bensoussan, A., Frehse, J.: Stochastic games with risk sensitive pay offs for N players. Le Matematiche 55(Suppl. 2), 5–54 (2000)
Borkar, V.S., Budhiraja, A.: Ergodic control for constrained diffusions: characterization using HJB equations. SIAM J. Control Optim. 43, 1467–1492 (2004/2005)
Borkar, V.S., Ghosh, M.K.: Stochastic differential games: occupation measure based approach. J. Optim. Theory Appl. 73, 359–385 (1992)
Bressan, A., Shen, W.: Small BV solutions of hyperbolic noncooperative differential games. SIAM J. Control Optim. 43, 194–215 (2004)
Capuzzo-Dolcetta, I., Lions, P.-L.: Hamilton-Jacobi equations with state constraints. Trans. A.M.S. 318, 643–683 (1990)
Cardaliaguet, P., Plaskacz, S.: Existence and uniqueness for a simple nonzero—sum differential game. Int. J. Game Theory 32, 33–71 (2003)
Chen, H., Mandelbaum, A.: Leontief systems, RBV’s and RBM’s. In: Davis, M.H.A., Elliott, R.J. (eds.) Proc. Imperial College workshop on Applied Stochastic Processes, pp. 1–43. Gordon and Breach, New York (1991)
Dupuis, P., Ishii, H.: SDE’s with oblique reflection on nonsmooth domains. Ann. Probab. 21, 554–580 (1993)
Fleming, W., Soner, H.: Controlled Makrov Processes and Viscosity Solutions. Springer, New York (1993)
Harrison, J.M., Reiman, M.I.: Reflected Brownian motion on an orthant. Ann. Probab. 9, 302–308 (1981)
Leitmann, G.: Cooperative and Non-Cooperative Many Players Differential Games. Springer, Wien (1974)
Manucci, P.: Nonzero sum stochastic differential games with discontinuous feedback. SIAM J. Control Optim. 43, 1222–1233 (2004/2005)
Mandelbaum, A., Pats, G.: State-dependent stochastic networks. Part I: approximations and applications with continuous diffusion limits. Ann. Appl. Probab. 8, 569–646 (1998)
Olsder, G.J.: On open and closed loop bang-bang control in nonzero—sum differential games. SIAM J. Control Optim. 40, 1087–1106 (2001/2002)
Ramasubramanian, S.: A subsidy-surplus model and the Skorokhod problem in an orthant. Math. Oper. Res. 25, 509–538 (2000)
Ramasubramanian, S.: An insurance network: Nash equilibrium. Insur. Math. Econ. 38, 374–390 (2006)
Reiman, M.I.: Open queueing networks in heavy traffic. Math. Oper. Res. 9, 441–458 (1984)
Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley, New York (2001)
Shashiashvili, M.: A lemma of variational distance between maximal functions with application to the Skorokhod problem in a nonnegative orthant with state dependent reflection directions. Stoch. Stoch. Rep. 48, 161–194 (1994)
Soner, H.M.: Optimal control with state space constraints I. SIAM J. Control Optim. 24, 552–561 (1986)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ramasubramanian, S. A d-person Differential Game with State Space Constraints. Appl Math Optim 56, 312–342 (2007). https://doi.org/10.1007/s00245-007-9011-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-007-9011-z