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Nonzero-sum stochastic differential games with additive structure and average payoffs
1. | Engineering Faculty, Universidad Veracruzana, Coatzacoalcos, Ver. 96538, Mexico |
2. | Faculty of Actuarial Sciences, Universidad Anáhuac, Av. Universidad Anáhuac 46. Col. Lomas Anáhuac, Huixquilucan, Edo. de México, Mexico |
References:
[1] |
R. Adams, Sobolev Spaces,, Academic Press, (1975).
|
[2] |
R. Akella and P. Kumar, Optimal control of production rate in a failure prone manufacturing system,, IEEE Trans. Automatic Control, 31 (1986), 116.
doi: 10.1109/TAC.1986.1104206. |
[3] |
A. Arapostathis and V. Borkar, Uniform recurrence properties of controlled diffusions and applications to optimal control,, SIAM J. Control Optim, 48 (2010), 4181.
doi: 10.1137/090762464. |
[4] |
E. Balder, Lectures on Young Measures,, Cahier de Mathématiques de la Décision, (1995). Google Scholar |
[5] |
M. Bardi, Explicit solutions of some linear-quadratic mean field games,, Netw. Heterog. Media., 7 (2012), 243.
doi: 10.3934/nhm.2012.7.243. |
[6] |
M. Bardi and F. Priuli, LQG Mean-Field Games with ergodic cost,, Proc. 52nd IEEE Conference on Decision and Control, (2013), 2493.
doi: 10.1109/CDC.2013.6760255. |
[7] |
A. Bhatt and R. Karandikar, Invariant measures and evolution equations for Markov processes,, Ann. Probab., 21 (1993), 2246.
doi: 10.1214/aop/1176989019. |
[8] |
P. Billingsley, Probability and Measure,, $3^{rd}$ edition, (1995).
|
[9] |
V. Bogachev, N. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions,, Comm. PartialDifferential Equations, 26 (2001), 2037.
doi: 10.1081/PDE-100107815. |
[10] |
V. Bogachev, N. Krylov and M. Röckner, Regularity and global bounds for the densities of invariant measures of diffusion processes,, Dokl. Akad. Nauk, 405 (2005), 583.
|
[11] |
V. Bogachev, M. Röckner and V. Stannat, Uniqueness of solutions of elliptic equations and uniqueness pf invariant measures of diffusions processes,, Sb. Math., 193 (2002), 945.
doi: 10.1070/SM2002v193n07ABEH000665. |
[12] |
V. Borkar, A topology for Markov controls,, Appl. Math. Optim., 20 (1989), 55.
doi: 10.1007/BF01447645. |
[13] |
V. Borkar and M. Ghosh, Stochastic differential games: Occupation measure based approach,, J. Optim. Theory Appl., 73 (1992), 359.
doi: 10.1007/BF00940187. |
[14] |
R. Buckdahn, P. Cardaliaguet and C. Rainer, Nash equilibrium payoffs for nonzero-sum stochastic diferential games,, SIAM J. Control Optim., 43 (2004), 624.
doi: 10.1137/S0363012902411556. |
[15] |
A. Calderón and J. Rosenblueth, Minimizing Approximate Original Solutions for Commensurate Delayed Controls,, Appl. Math. Lett., 7 (1994), 5.
doi: 10.1016/0893-9659(94)90063-9. |
[16] |
R. Durrett, Stochastic Calculus: A Practical Introduction,, CRC Press, (1996).
|
[17] |
A. Dvoretzky, A. Wald and J. Wolfowitz, Elimination of randomization in certain statistical decision procedure and zero-sum two person games,, Ann. Math. Statist., 22 (1951), 1.
doi: 10.1214/aoms/1177729689. |
[18] |
B. Escobedo-Trujillo, J. López-Barrientos and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum stochastic differential games,, J. Optim. Theory Appl., 153 (2012), 662.
doi: 10.1007/s10957-011-9974-4. |
[19] |
K. Fan, Fixed-point and minimax theorems in locally convex linear spaces,, Proc. N.A.S.U.S.A., 38 (1952), 121.
doi: 10.1073/pnas.38.2.121. |
[20] |
K. Fan, Minimax theorems,, Proc. N.A.S.U.S.A., 39 (1953), 32.
doi: 10.1073/pnas.39.1.42. |
[21] |
G. Folland, Real Analysis. Modern Techniques and Their Applications,, $2^{nd}$ edition, (1999).
|
[22] |
A. Friedman, Stochastic Differential Equations and Applications,, Vol. 1, (1975).
|
[23] |
M. Ghosh, A. Arapostathis and S. Marcus, Ergodic control of switching diffusions,, SIAM J. Control Optim., 35 (1997), 1962.
doi: 10.1137/S0363012996299302. |
[24] |
M. Ghosh and A. Bagchi, Stochastic game with average payoff criterion,, Appl. Math. Optim., 38 (1998), 283.
doi: 10.1007/s002459900092. |
[25] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprinted version, (1998). Google Scholar |
[26] |
P. Glynn and S. Meyn, A Liapounov bound for solutions of the Poisson equation,, Ann. Prob., 24 (1996), 916.
doi: 10.1214/aop/1039639370. |
[27] |
J. González-Hernández and O. Hernández-Lerma, Extreme points of sets of randomized strategies in constrained optimization and control problems,, SIAM J. Optim., 15 (2005), 1085.
doi: 10.1137/040605345. |
[28] |
H. Hernández-Hernández, Existence of nash equilibria in discounted nonzero-sum stochastic games with additive structure,, Morfismos, 6 (2002), 43. Google Scholar |
[29] |
O. Hernández-Lerma and J. Lasserre, Discrete-Time Markov Control Processes: Basic Optimality Criteria,, Springer, (1996).
doi: 10.1007/978-1-4612-0729-0. |
[30] |
O. Hernández-Lerma and J. Lasserre, Further topics on Discrete-Time Markov Control Processes,, Springer, (1999).
doi: 10.1007/978-1-4612-0561-6. |
[31] |
H. Jasso-Fuentes and O. Hernández-Lerma, Characterizations of overtaking optimality for controlled diffusion processes,, Appl. Math. Optim., 57 (2008), 349.
doi: 10.1007/s00245-007-9025-6. |
[32] |
F. Klebaner, Introduction to Stochastic Calculus with Applications,, $2^{nd}$ edition, (2005).
doi: 10.1142/p386. |
[33] |
A. Knapp, Advanced Real Analysis,, $2^{nd}$ edition, (2005).
|
[34] |
A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis,, Dover Publications, (1999). Google Scholar |
[35] |
N. Krylov, Controlled diffusion processes. Applications of Mathematics,, Springer-Verlag, (1980).
|
[36] |
H. Kushner, Numerical approximations for nonzero-sum stochastic differential games,, SIAM J.Control Optim., 46 (2007), 1942.
doi: 10.1137/050647931. |
[37] |
H. Küenle, Equilibrium strategies in stochastic games with additive cost and transition structure,, Internat. Game Theory Rev., 1 (1999), 131.
doi: 10.1142/S0219198999000098. |
[38] |
S. Meyn and R. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time precesses,, Adv. Appl. Prob., 25 (1993), 518.
doi: 10.2307/1427522. |
[39] |
H. Morimoto and M. Ohashi, On linear stochastic differential games with average cost criterions,, J. Optimization Theory and Applications, 64 (1990), 127.
doi: 10.1007/BF00940027. |
[40] |
J. Munkres, Topology,, $2^{nd}$ edition, (2000). Google Scholar |
[41] |
A. Nowak, Stationary equilibria for nonzero-sum average payoff ergodic stochastic games with general state space,, Advances in Dynamic Games and Applications (T. Basar and A. Haurie, 1 (1994), 231.
|
[42] |
A. Nowak, Remark on sensitive equilibria in stochastic games with additive reward and transition structure,, Math. Meth. Oper. Res., 64 (2006), 481.
doi: 10.1007/s00186-006-0090-4. |
[43] |
P. Pedregal, Optimization, relaxation and Young measures,, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 27.
doi: 10.1090/S0273-0979-99-00774-0. |
[44] |
B. Oksendal, Stochastic Differential Equations: An Introduction with Applications,, $4^{th}$ edition, (1995).
doi: 10.1007/978-3-662-03185-8. |
[45] |
K. Parthasarathy and T. Steerneman, A tool in establishing total variation convergence,, Proceedings of the American Mathematical Society, 95 (1985), 626.
doi: 10.1090/S0002-9939-1985-0810175-X. |
[46] |
T. Prieto-Rumeau and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games,, Math. Meth. Oper. Res., 61 (2005), 437.
doi: 10.1007/s001860400392. |
[47] |
T. Raghavan, S. Tijs and O. Vrieze, On stochastic games with additive reward and transition structure,, J. Optim. Theory Appl., 47 (1985), 451.
doi: 10.1007/BF00942191. |
[48] |
J. Rosenblueth, Proper relaxation of optimal control problem,, J. Optim. Theory Appl., 74 (1992), 509.
doi: 10.1007/BF00940324. |
[49] |
N. Saldi, T. Linder and S. Yüksel, Asymptotic optimality of quantized control in Markov decision processes., Submitted to IEEE Conference on Decision and Control., (). Google Scholar |
[50] |
M. Schäl, Conditions for optimality and for the limit of $n-$stage optimal policies to be optimal,, Z. Wahrs. Verw. Gerb., 32 (1975), 179.
doi: 10.1007/BF00532612. |
[51] |
L. Young, Lectures on the Calculus of Variations and Optimal Control Theory,, W. B. Saunders, (1969).
|
[52] |
J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972).
|
[53] |
E. Zeidler, Nonlinear Functional Analysis and its Applications II/A,, Springer-Verlag, (1990).
doi: 10.1007/978-1-4612-0985-0. |
show all references
References:
[1] |
R. Adams, Sobolev Spaces,, Academic Press, (1975).
|
[2] |
R. Akella and P. Kumar, Optimal control of production rate in a failure prone manufacturing system,, IEEE Trans. Automatic Control, 31 (1986), 116.
doi: 10.1109/TAC.1986.1104206. |
[3] |
A. Arapostathis and V. Borkar, Uniform recurrence properties of controlled diffusions and applications to optimal control,, SIAM J. Control Optim, 48 (2010), 4181.
doi: 10.1137/090762464. |
[4] |
E. Balder, Lectures on Young Measures,, Cahier de Mathématiques de la Décision, (1995). Google Scholar |
[5] |
M. Bardi, Explicit solutions of some linear-quadratic mean field games,, Netw. Heterog. Media., 7 (2012), 243.
doi: 10.3934/nhm.2012.7.243. |
[6] |
M. Bardi and F. Priuli, LQG Mean-Field Games with ergodic cost,, Proc. 52nd IEEE Conference on Decision and Control, (2013), 2493.
doi: 10.1109/CDC.2013.6760255. |
[7] |
A. Bhatt and R. Karandikar, Invariant measures and evolution equations for Markov processes,, Ann. Probab., 21 (1993), 2246.
doi: 10.1214/aop/1176989019. |
[8] |
P. Billingsley, Probability and Measure,, $3^{rd}$ edition, (1995).
|
[9] |
V. Bogachev, N. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions,, Comm. PartialDifferential Equations, 26 (2001), 2037.
doi: 10.1081/PDE-100107815. |
[10] |
V. Bogachev, N. Krylov and M. Röckner, Regularity and global bounds for the densities of invariant measures of diffusion processes,, Dokl. Akad. Nauk, 405 (2005), 583.
|
[11] |
V. Bogachev, M. Röckner and V. Stannat, Uniqueness of solutions of elliptic equations and uniqueness pf invariant measures of diffusions processes,, Sb. Math., 193 (2002), 945.
doi: 10.1070/SM2002v193n07ABEH000665. |
[12] |
V. Borkar, A topology for Markov controls,, Appl. Math. Optim., 20 (1989), 55.
doi: 10.1007/BF01447645. |
[13] |
V. Borkar and M. Ghosh, Stochastic differential games: Occupation measure based approach,, J. Optim. Theory Appl., 73 (1992), 359.
doi: 10.1007/BF00940187. |
[14] |
R. Buckdahn, P. Cardaliaguet and C. Rainer, Nash equilibrium payoffs for nonzero-sum stochastic diferential games,, SIAM J. Control Optim., 43 (2004), 624.
doi: 10.1137/S0363012902411556. |
[15] |
A. Calderón and J. Rosenblueth, Minimizing Approximate Original Solutions for Commensurate Delayed Controls,, Appl. Math. Lett., 7 (1994), 5.
doi: 10.1016/0893-9659(94)90063-9. |
[16] |
R. Durrett, Stochastic Calculus: A Practical Introduction,, CRC Press, (1996).
|
[17] |
A. Dvoretzky, A. Wald and J. Wolfowitz, Elimination of randomization in certain statistical decision procedure and zero-sum two person games,, Ann. Math. Statist., 22 (1951), 1.
doi: 10.1214/aoms/1177729689. |
[18] |
B. Escobedo-Trujillo, J. López-Barrientos and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum stochastic differential games,, J. Optim. Theory Appl., 153 (2012), 662.
doi: 10.1007/s10957-011-9974-4. |
[19] |
K. Fan, Fixed-point and minimax theorems in locally convex linear spaces,, Proc. N.A.S.U.S.A., 38 (1952), 121.
doi: 10.1073/pnas.38.2.121. |
[20] |
K. Fan, Minimax theorems,, Proc. N.A.S.U.S.A., 39 (1953), 32.
doi: 10.1073/pnas.39.1.42. |
[21] |
G. Folland, Real Analysis. Modern Techniques and Their Applications,, $2^{nd}$ edition, (1999).
|
[22] |
A. Friedman, Stochastic Differential Equations and Applications,, Vol. 1, (1975).
|
[23] |
M. Ghosh, A. Arapostathis and S. Marcus, Ergodic control of switching diffusions,, SIAM J. Control Optim., 35 (1997), 1962.
doi: 10.1137/S0363012996299302. |
[24] |
M. Ghosh and A. Bagchi, Stochastic game with average payoff criterion,, Appl. Math. Optim., 38 (1998), 283.
doi: 10.1007/s002459900092. |
[25] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprinted version, (1998). Google Scholar |
[26] |
P. Glynn and S. Meyn, A Liapounov bound for solutions of the Poisson equation,, Ann. Prob., 24 (1996), 916.
doi: 10.1214/aop/1039639370. |
[27] |
J. González-Hernández and O. Hernández-Lerma, Extreme points of sets of randomized strategies in constrained optimization and control problems,, SIAM J. Optim., 15 (2005), 1085.
doi: 10.1137/040605345. |
[28] |
H. Hernández-Hernández, Existence of nash equilibria in discounted nonzero-sum stochastic games with additive structure,, Morfismos, 6 (2002), 43. Google Scholar |
[29] |
O. Hernández-Lerma and J. Lasserre, Discrete-Time Markov Control Processes: Basic Optimality Criteria,, Springer, (1996).
doi: 10.1007/978-1-4612-0729-0. |
[30] |
O. Hernández-Lerma and J. Lasserre, Further topics on Discrete-Time Markov Control Processes,, Springer, (1999).
doi: 10.1007/978-1-4612-0561-6. |
[31] |
H. Jasso-Fuentes and O. Hernández-Lerma, Characterizations of overtaking optimality for controlled diffusion processes,, Appl. Math. Optim., 57 (2008), 349.
doi: 10.1007/s00245-007-9025-6. |
[32] |
F. Klebaner, Introduction to Stochastic Calculus with Applications,, $2^{nd}$ edition, (2005).
doi: 10.1142/p386. |
[33] |
A. Knapp, Advanced Real Analysis,, $2^{nd}$ edition, (2005).
|
[34] |
A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis,, Dover Publications, (1999). Google Scholar |
[35] |
N. Krylov, Controlled diffusion processes. Applications of Mathematics,, Springer-Verlag, (1980).
|
[36] |
H. Kushner, Numerical approximations for nonzero-sum stochastic differential games,, SIAM J.Control Optim., 46 (2007), 1942.
doi: 10.1137/050647931. |
[37] |
H. Küenle, Equilibrium strategies in stochastic games with additive cost and transition structure,, Internat. Game Theory Rev., 1 (1999), 131.
doi: 10.1142/S0219198999000098. |
[38] |
S. Meyn and R. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time precesses,, Adv. Appl. Prob., 25 (1993), 518.
doi: 10.2307/1427522. |
[39] |
H. Morimoto and M. Ohashi, On linear stochastic differential games with average cost criterions,, J. Optimization Theory and Applications, 64 (1990), 127.
doi: 10.1007/BF00940027. |
[40] |
J. Munkres, Topology,, $2^{nd}$ edition, (2000). Google Scholar |
[41] |
A. Nowak, Stationary equilibria for nonzero-sum average payoff ergodic stochastic games with general state space,, Advances in Dynamic Games and Applications (T. Basar and A. Haurie, 1 (1994), 231.
|
[42] |
A. Nowak, Remark on sensitive equilibria in stochastic games with additive reward and transition structure,, Math. Meth. Oper. Res., 64 (2006), 481.
doi: 10.1007/s00186-006-0090-4. |
[43] |
P. Pedregal, Optimization, relaxation and Young measures,, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 27.
doi: 10.1090/S0273-0979-99-00774-0. |
[44] |
B. Oksendal, Stochastic Differential Equations: An Introduction with Applications,, $4^{th}$ edition, (1995).
doi: 10.1007/978-3-662-03185-8. |
[45] |
K. Parthasarathy and T. Steerneman, A tool in establishing total variation convergence,, Proceedings of the American Mathematical Society, 95 (1985), 626.
doi: 10.1090/S0002-9939-1985-0810175-X. |
[46] |
T. Prieto-Rumeau and O. Hernández-Lerma, Bias and overtaking equilibria for zero-sum continuous-time Markov games,, Math. Meth. Oper. Res., 61 (2005), 437.
doi: 10.1007/s001860400392. |
[47] |
T. Raghavan, S. Tijs and O. Vrieze, On stochastic games with additive reward and transition structure,, J. Optim. Theory Appl., 47 (1985), 451.
doi: 10.1007/BF00942191. |
[48] |
J. Rosenblueth, Proper relaxation of optimal control problem,, J. Optim. Theory Appl., 74 (1992), 509.
doi: 10.1007/BF00940324. |
[49] |
N. Saldi, T. Linder and S. Yüksel, Asymptotic optimality of quantized control in Markov decision processes., Submitted to IEEE Conference on Decision and Control., (). Google Scholar |
[50] |
M. Schäl, Conditions for optimality and for the limit of $n-$stage optimal policies to be optimal,, Z. Wahrs. Verw. Gerb., 32 (1975), 179.
doi: 10.1007/BF00532612. |
[51] |
L. Young, Lectures on the Calculus of Variations and Optimal Control Theory,, W. B. Saunders, (1969).
|
[52] |
J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972).
|
[53] |
E. Zeidler, Nonlinear Functional Analysis and its Applications II/A,, Springer-Verlag, (1990).
doi: 10.1007/978-1-4612-0985-0. |
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