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doi: 10.3934/mcrf.2021025
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A nonzero-sum risk-sensitive stochastic differential game in the orthant

1. 

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

2. 

Department of Mathematics, Indian Institute of Science Education and Research, Pune, Maharashtra 411008, India

* Corresponding author: Somnath Pradhan

Received  July 2020 Revised  January 2021 Early access April 2021

Fund Project: This work is partially supported by UGC Center for Advanced Study

We study a nonzero-sum risk-sensitive stochastic differential game for controlled reflecting diffusion processes in the nonnegative orthant. We treat two cost evaluation criteria, namely, discounted cost and ergodic cost. Under certain assumptions, we establish the existence of Nash equilibria. Also, we completely characterize a Nash equilibrium for the ergodic cost criterion in the space of stationary Markov strategies.

Citation: Mrinal K. Ghosh, Somnath Pradhan. A nonzero-sum risk-sensitive stochastic differential game in the orthant. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021025
References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York 1975.  Google Scholar

[2]

A. Arapostathis and A. Biswas, Infinite horizon risk-sensitive control of diffusions without any blanket stability assumptions, Stochastic Process. Appl., 128 (2018), 1485-1524.  doi: 10.1016/j.spa.2017.08.001.  Google Scholar

[3]

A. ArapostathisA. BiswasV. S. Borkar and K. S. Kumar, A variational characterization of the risk-sensitive average reward for controlled diffusions on $\mathbb{R}^{d}$, SIAM J. Control Optim., 58 (2020), 3785-3813.  doi: 10.1137/20M1329202.  Google Scholar

[4]

A. Arapostathis, A. Biswas and S. Saha, Strict monotonicity of principal eigenvalues of elliptic operators in $\mathbb{R}^{d}$ and risk-sensitive control, J. Math. Pures Appl., 124 (2019), 169–219. doi: 10.1016/j.matpur.2018.05.008.  Google Scholar

[5]

A. Arapostathis, V. S. Borkar and M. K. Ghosh, Ergodic Control of Diffusion Processes, Cambridge University Press, Cambridge, U.K. 2012.  Google Scholar

[6]

A. ArapostathisV. S. Borkar and K. S. Kumar, Risk-sensitive control and an abstract Collatz-Wielandt formula, J. Theoret. Probab., 29 (2016), 1458-1484.  doi: 10.1007/s10959-015-0616-x.  Google Scholar

[7]

A. Bagchi and K. Suresh Kumar, Dynamic asset management with risk-sensitive criterion and non-negative factor constraints: A differential game approach, Stochastics, 81 (2009), 503-530.  doi: 10.1080/17442500902774917.  Google Scholar

[8]

A. Basu and M. K. Ghosh, Zero-sum risk-sensitive stochastic differential games, Math. Oper. Res., 37 (2012), 437-449.  doi: 10.1287/moor.1120.0542.  Google Scholar

[9]

V. E. Bene$\breve{s}$, Existence of optimal strategies based on specified information of a class of stochastic decision problems, SIAM J. Control, 8 (1970), 179-188.  doi: 10.1137/0308012.  Google Scholar

[10]

A. Biswas, An eigenvalue approach to the risk-sensitive control problem in near monotone case, Systems Control Lett., 60 (2011), 181-184.  doi: 10.1016/j.sysconle.2010.12.002.  Google Scholar

[11]

A. Biswas, V. S. Borkar and K. Suresh Kumar, Risk-sensitive control with near monotone cost, Appl. Math. Optim., 62 (2009), 145–163. Errata corriege Ⅰ. ibid 62 (2010), 165–167. Errata corriege Ⅱ. ibid 62 (2010), 435–438. doi: 10.1007/s00245-010-9119-4.  Google Scholar

[12]

A. Biswas and S. Saha, Zero-sum stochastic differential games with risk-sensitive cost, App. Math. Optim., 81 (2020), 113-140.  doi: 10.1007/s00245-018-9479-8.  Google Scholar

[13]

V. Borkar and A. Budhiraja, Ergodic control for constained diffusions: Characterization using HJB equation, SIAM J. Control Optim., 43 (2005), 1467-1492.  doi: 10.1137/S0363012902417619.  Google Scholar

[14]

V. S. Borkar and M. K. Ghosh, Stochastic differential games: occupation measure based approach, J. Optim. Theory Appl., 73 (1992), 359-385. Errata corriege. ibid 88 (1996), 251–252. doi: 10.1007/BF02192034.  Google Scholar

[15]

A. Budhiraja, An ergodic control problem for constrained diffusion processes: Existence of optimal Markov control, SIAM J. Control Optim., 42 (2003), 532-558.  doi: 10.1137/S0363012901379073.  Google Scholar

[16]

K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger's Equation, Grundlehren der Mathematischen Wissenschaften, 312, Springer, Berlin, 1995. doi: 10.1007/978-3-642-57856-4.  Google Scholar

[17]

K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121.  Google Scholar

[18]

J. Filar and K. Vrieze, Competitive Markov Decision Processes, Springer-Verlag, New York, 1997.  Google Scholar

[19]

S. K. GauttamK. S. Kumar and C. Pal, Risk-sensitive control of reflected diffusion process on orthrant, Pure Appl. Funct. Anal., 2 (2017), 477-510.   Google Scholar

[20]

M. K. Ghosh and A. Bagchi, Stochastic games with average payoff criterion, Appl. Math. Optim., 38 (1998), 283-301.  doi: 10.1007/s002459900092.  Google Scholar

[21]

M. K. Ghosh and S. Pradhan, Risk-sensitive stochastic differential game with reflecting diffusions, Stoch. Anal. Appl., 36 (2018), 1-27.  doi: 10.1080/07362994.2017.1356732.  Google Scholar

[22]

M. K. Ghosh and S. Pradhan, Zero-sum risk-sensitive stochastic differential games with reflecting diffusions in the orthant, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 114, 33 pp. doi: 10.1051/cocv/2020029.  Google Scholar

[23]

M. K. Ghosh and K. Suresh Kumar, A stochastic differential game in the orthant, J. Math. Anal. Appl., 265 (2002), 12-37.  doi: 10.1006/jmaa.2001.7679.  Google Scholar

[24]

M. K. Ghosh and K. Suresh Kumar, A nonzero-sum stochastic differential game in the orthant, J. Math. Anal. Appl., 305 (2005), 158-174.  doi: 10.1016/j.jmaa.2004.11.002.  Google Scholar

[25]

M. K. Ghosh and K. Suresh Kumar, Nonzero-sum risk-sensitive stochastic differential games with reflecting diffusions, Comput. Appl. Math., 18 (1999), 355-368.   Google Scholar

[26]

M. K. Ghosh, K. Suresh Kumar and C. Pal, Nonzero-sum risk-sensitive stochastic differential games, arXiv: 1604.01142 Google Scholar

[27]

M. K. Ghosh, K. Suresh Kumar, C. Pal and S. Pradhan, Nonzero-sum risk-sensitive stochastic differential games with discounted costs, Stochastic Analysis and Applications, 39 (2021), 306-326. doi: 10.1080/07362994.2020.1796707.  Google Scholar

[28]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 224, 2nd ed., Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[29]

C. J. HimmelbergT. ParthasarathyT. E. S. Raghavan and F. S. Van Fleck, Existence of $p$-equilibrium and optimal stationary strategies in stochastic games, Proc. Amer. Math. Soc., 60 (1976), 245-251.  doi: 10.2307/2041151.  Google Scholar

[30]

D. L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic, Adv. in Appl. Prob., 2 (1970), 150-177.  doi: 10.2307/3518347.  Google Scholar

[31]

H. J. Kushner and L. F. Martins, Limit theorems for pathwise average cost per unit time problems for controlled queues in heavy traffic, Stochastics Stochastics Rep., 42 (1993), 25-51.  doi: 10.1080/17442509308833808.  Google Scholar

[32]

H. J. Kushner and K. M. Ramachandran, Optimal and approximately optimal control policies for queues in heavy traffic, SIAM J. Control Optim., 27 (1989), 1293–1318. doi: 10.1137/0327066.  Google Scholar

[33]

A. J. Lemoine, Network of queues - A survey of weak convergence results, Management Science, 24 (1978), 1175-1193.  doi: 10.1287/mnsc.24.11.1175.  Google Scholar

[34]

G. Leoni, A First Course in Sobolev Spaces, Graduate Studies in Mathematics, 105, AMS, Rhode Island USA, 2009. doi: 10.1090/gsm/105.  Google Scholar

[35]

G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.  Google Scholar

[36]

G. M. Lieberman, Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations, Trans. Amer. Math. Soc., 304 (1987), 343-353.  doi: 10.1090/S0002-9947-1987-0906819-0.  Google Scholar

[37]

P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 37 (1984), 511-537.  doi: 10.1002/cpa.3160370408.  Google Scholar

[38]

J.-L. Menaldi and M. Robin, Remarks on risk-sensitive control problems, Appl. Math. Optim., 52 (2005), 297-310.  doi: 10.1007/s00245-005-0829-y.  Google Scholar

[39]

A. S. Nowak, Notes on risk-sensitive Nash equilibria. Advances in dynamic games, Ann. International. Soc. Dynam. Games, 7 (2005), 95-109.  doi: 10.1007/0-8176-4429-6_5.  Google Scholar

[40]

S. Pradhan, Risk-sensitive ergodic control of reflected diffusion processes in orthant, Appl. Math. Optim., (2019), to appear. doi: 10.1007/s00245-019-09606-w.  Google Scholar

[41]

M. I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res., 9 (1984), 441-458.  doi: 10.1287/moor.9.3.441.  Google Scholar

[42]

A. Yu. Veretennikov, On strong and weak solutions of one-dimensional stochastic equations with boundary conditions, Theory Probab. Appl., 26 (1981), 670-686.   Google Scholar

[43]

J. Warga, Functions of relaxed controls, SIAM J. Control, 5 (1967), 628-641.  doi: 10.1137/0305042.  Google Scholar

[44]

Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations, World Scientific, Singapore, 2006. doi: 10.1142/6238.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York 1975.  Google Scholar

[2]

A. Arapostathis and A. Biswas, Infinite horizon risk-sensitive control of diffusions without any blanket stability assumptions, Stochastic Process. Appl., 128 (2018), 1485-1524.  doi: 10.1016/j.spa.2017.08.001.  Google Scholar

[3]

A. ArapostathisA. BiswasV. S. Borkar and K. S. Kumar, A variational characterization of the risk-sensitive average reward for controlled diffusions on $\mathbb{R}^{d}$, SIAM J. Control Optim., 58 (2020), 3785-3813.  doi: 10.1137/20M1329202.  Google Scholar

[4]

A. Arapostathis, A. Biswas and S. Saha, Strict monotonicity of principal eigenvalues of elliptic operators in $\mathbb{R}^{d}$ and risk-sensitive control, J. Math. Pures Appl., 124 (2019), 169–219. doi: 10.1016/j.matpur.2018.05.008.  Google Scholar

[5]

A. Arapostathis, V. S. Borkar and M. K. Ghosh, Ergodic Control of Diffusion Processes, Cambridge University Press, Cambridge, U.K. 2012.  Google Scholar

[6]

A. ArapostathisV. S. Borkar and K. S. Kumar, Risk-sensitive control and an abstract Collatz-Wielandt formula, J. Theoret. Probab., 29 (2016), 1458-1484.  doi: 10.1007/s10959-015-0616-x.  Google Scholar

[7]

A. Bagchi and K. Suresh Kumar, Dynamic asset management with risk-sensitive criterion and non-negative factor constraints: A differential game approach, Stochastics, 81 (2009), 503-530.  doi: 10.1080/17442500902774917.  Google Scholar

[8]

A. Basu and M. K. Ghosh, Zero-sum risk-sensitive stochastic differential games, Math. Oper. Res., 37 (2012), 437-449.  doi: 10.1287/moor.1120.0542.  Google Scholar

[9]

V. E. Bene$\breve{s}$, Existence of optimal strategies based on specified information of a class of stochastic decision problems, SIAM J. Control, 8 (1970), 179-188.  doi: 10.1137/0308012.  Google Scholar

[10]

A. Biswas, An eigenvalue approach to the risk-sensitive control problem in near monotone case, Systems Control Lett., 60 (2011), 181-184.  doi: 10.1016/j.sysconle.2010.12.002.  Google Scholar

[11]

A. Biswas, V. S. Borkar and K. Suresh Kumar, Risk-sensitive control with near monotone cost, Appl. Math. Optim., 62 (2009), 145–163. Errata corriege Ⅰ. ibid 62 (2010), 165–167. Errata corriege Ⅱ. ibid 62 (2010), 435–438. doi: 10.1007/s00245-010-9119-4.  Google Scholar

[12]

A. Biswas and S. Saha, Zero-sum stochastic differential games with risk-sensitive cost, App. Math. Optim., 81 (2020), 113-140.  doi: 10.1007/s00245-018-9479-8.  Google Scholar

[13]

V. Borkar and A. Budhiraja, Ergodic control for constained diffusions: Characterization using HJB equation, SIAM J. Control Optim., 43 (2005), 1467-1492.  doi: 10.1137/S0363012902417619.  Google Scholar

[14]

V. S. Borkar and M. K. Ghosh, Stochastic differential games: occupation measure based approach, J. Optim. Theory Appl., 73 (1992), 359-385. Errata corriege. ibid 88 (1996), 251–252. doi: 10.1007/BF02192034.  Google Scholar

[15]

A. Budhiraja, An ergodic control problem for constrained diffusion processes: Existence of optimal Markov control, SIAM J. Control Optim., 42 (2003), 532-558.  doi: 10.1137/S0363012901379073.  Google Scholar

[16]

K. L. Chung and Z. X. Zhao, From Brownian Motion to Schrödinger's Equation, Grundlehren der Mathematischen Wissenschaften, 312, Springer, Berlin, 1995. doi: 10.1007/978-3-642-57856-4.  Google Scholar

[17]

K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121.  Google Scholar

[18]

J. Filar and K. Vrieze, Competitive Markov Decision Processes, Springer-Verlag, New York, 1997.  Google Scholar

[19]

S. K. GauttamK. S. Kumar and C. Pal, Risk-sensitive control of reflected diffusion process on orthrant, Pure Appl. Funct. Anal., 2 (2017), 477-510.   Google Scholar

[20]

M. K. Ghosh and A. Bagchi, Stochastic games with average payoff criterion, Appl. Math. Optim., 38 (1998), 283-301.  doi: 10.1007/s002459900092.  Google Scholar

[21]

M. K. Ghosh and S. Pradhan, Risk-sensitive stochastic differential game with reflecting diffusions, Stoch. Anal. Appl., 36 (2018), 1-27.  doi: 10.1080/07362994.2017.1356732.  Google Scholar

[22]

M. K. Ghosh and S. Pradhan, Zero-sum risk-sensitive stochastic differential games with reflecting diffusions in the orthant, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 114, 33 pp. doi: 10.1051/cocv/2020029.  Google Scholar

[23]

M. K. Ghosh and K. Suresh Kumar, A stochastic differential game in the orthant, J. Math. Anal. Appl., 265 (2002), 12-37.  doi: 10.1006/jmaa.2001.7679.  Google Scholar

[24]

M. K. Ghosh and K. Suresh Kumar, A nonzero-sum stochastic differential game in the orthant, J. Math. Anal. Appl., 305 (2005), 158-174.  doi: 10.1016/j.jmaa.2004.11.002.  Google Scholar

[25]

M. K. Ghosh and K. Suresh Kumar, Nonzero-sum risk-sensitive stochastic differential games with reflecting diffusions, Comput. Appl. Math., 18 (1999), 355-368.   Google Scholar

[26]

M. K. Ghosh, K. Suresh Kumar and C. Pal, Nonzero-sum risk-sensitive stochastic differential games, arXiv: 1604.01142 Google Scholar

[27]

M. K. Ghosh, K. Suresh Kumar, C. Pal and S. Pradhan, Nonzero-sum risk-sensitive stochastic differential games with discounted costs, Stochastic Analysis and Applications, 39 (2021), 306-326. doi: 10.1080/07362994.2020.1796707.  Google Scholar

[28]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 224, 2nd ed., Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[29]

C. J. HimmelbergT. ParthasarathyT. E. S. Raghavan and F. S. Van Fleck, Existence of $p$-equilibrium and optimal stationary strategies in stochastic games, Proc. Amer. Math. Soc., 60 (1976), 245-251.  doi: 10.2307/2041151.  Google Scholar

[30]

D. L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic, Adv. in Appl. Prob., 2 (1970), 150-177.  doi: 10.2307/3518347.  Google Scholar

[31]

H. J. Kushner and L. F. Martins, Limit theorems for pathwise average cost per unit time problems for controlled queues in heavy traffic, Stochastics Stochastics Rep., 42 (1993), 25-51.  doi: 10.1080/17442509308833808.  Google Scholar

[32]

H. J. Kushner and K. M. Ramachandran, Optimal and approximately optimal control policies for queues in heavy traffic, SIAM J. Control Optim., 27 (1989), 1293–1318. doi: 10.1137/0327066.  Google Scholar

[33]

A. J. Lemoine, Network of queues - A survey of weak convergence results, Management Science, 24 (1978), 1175-1193.  doi: 10.1287/mnsc.24.11.1175.  Google Scholar

[34]

G. Leoni, A First Course in Sobolev Spaces, Graduate Studies in Mathematics, 105, AMS, Rhode Island USA, 2009. doi: 10.1090/gsm/105.  Google Scholar

[35]

G. M. Lieberman, Oblique Derivative Problems for Elliptic Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8679.  Google Scholar

[36]

G. M. Lieberman, Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations, Trans. Amer. Math. Soc., 304 (1987), 343-353.  doi: 10.1090/S0002-9947-1987-0906819-0.  Google Scholar

[37]

P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 37 (1984), 511-537.  doi: 10.1002/cpa.3160370408.  Google Scholar

[38]

J.-L. Menaldi and M. Robin, Remarks on risk-sensitive control problems, Appl. Math. Optim., 52 (2005), 297-310.  doi: 10.1007/s00245-005-0829-y.  Google Scholar

[39]

A. S. Nowak, Notes on risk-sensitive Nash equilibria. Advances in dynamic games, Ann. International. Soc. Dynam. Games, 7 (2005), 95-109.  doi: 10.1007/0-8176-4429-6_5.  Google Scholar

[40]

S. Pradhan, Risk-sensitive ergodic control of reflected diffusion processes in orthant, Appl. Math. Optim., (2019), to appear. doi: 10.1007/s00245-019-09606-w.  Google Scholar

[41]

M. I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res., 9 (1984), 441-458.  doi: 10.1287/moor.9.3.441.  Google Scholar

[42]

A. Yu. Veretennikov, On strong and weak solutions of one-dimensional stochastic equations with boundary conditions, Theory Probab. Appl., 26 (1981), 670-686.   Google Scholar

[43]

J. Warga, Functions of relaxed controls, SIAM J. Control, 5 (1967), 628-641.  doi: 10.1137/0305042.  Google Scholar

[44]

Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations, World Scientific, Singapore, 2006. doi: 10.1142/6238.  Google Scholar

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