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An active-set strategy to solve Markov decision processes with good-deal risk measure

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Abstract

This paper proposes a quasi closed-form solution for the reweighting of transition probabilities in finite state, finite action distributionally robust Markov decision processes with good-deal risk measure. The relation to the expected (risk-neutral) and minimax (worst-case) discounted cumulated cost objectives is discussed, as well as possible methods for the choice of the risk measure parameters. Numerical results illustrate the computational effectiveness of the proposed approach.

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References

  1. Abada, I., Ehrenmann, A., Smeers, Y.: Modeling gas markets with endogenous long-term contracts. Oper. Res. 65(4), 856–877 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Acerbi, C.: Spectral measures of risk: a coherent representation of subjective risk aversion. J. Bank. Finance 26(7), 1505–1518 (2002)

    Article  Google Scholar 

  3. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Prog. 95(1), 3–51 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Artzner, P., Delbaen, F., Eber, J.M., Heath, D., Ku, H.: Coherent multi-period risk adjusted values and Bellman’s principle. Ann. Oper. Res. 152(1), 5–22 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Becherer, D., Kentia, K.: Good deal hedging and valuation under combined uncertainty about drift and volatility. Probab. Uncertain. Quant. Risk 2(1), 13 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bellman, R.E., Dreyfus, S.E.: Applied Dynamic Programming. Princeton University Press, Princeton (2015)

    MATH  Google Scholar 

  7. Bertsekas, D.P., Tsitsiklis, J.N.: Neuro-dynamic programming: an overview. In: Proceedings of the 34th IEEE Conference on Decision and Control, vol. 1, pp. 560–564. IEEE (1995)

  8. Björk, T., Slinko, I.: Towards a general theory of good-deal bounds. Rev. Finance 10(2), 221–260 (2006)

    Article  MATH  Google Scholar 

  9. Chatterjee, K., Sen, K., Henzinger, T.A.: Model-checking \(\omega \)-regular properties of interval Markov chains. In: International Conference on Foundations of Software Science and Computational Structures, pp. 302–317. Springer (2008)

  10. Cheridito, P., Delbaen, F., Kupper, M., et al.: Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11, 57–106 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chung, K.J., Sobel, M.J.: Discounted MDPs: distribution functions and exponential utility maximization. SIAM J. Control Optim. 25(1), 49–62 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cochrane, J.H., Saa-Requejo, J.: Beyond arbitrage: good-deal asset price bounds in incomplete markets. J. Polit Econ 108(1), 79–119 (2000)

    Article  Google Scholar 

  13. Delage, E., Ye, Y.: Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3), 595–612 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Delbaen, F.: Coherent risk measures on general probability spaces. In: Sandmann K., Schönbucher P.J. (eds.) Advances in Finance and Stochastics. Springer, Berlin, Heidelberg (2002)

  15. Domahidi, A., Chu, E., Boyd, S.: ECOS: an SOCP solver for embedded systems. In: European Control Conference (ECC), pp. 3071–3076 (2013)

  16. Druenne, E., Ehrenmann, A., de Maere d’Aertrycke, G., Smeers, Y.: Good-deal investment valuation in stochastic generation capacity expansion problems. In: 44th Hawaii International Conference on System Sciences (HICSS), pp. 1–9. IEEE (2011)

  17. Epstein, L., Schneider, M.: Recursive multiple-priors. J. Econ. Theory 113(1), 1–31 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stoch. 6(4), 429–447 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Frittelli, M., Gianin, E.R.: Dynamic convex risk measures. In: Risk Measures for the 21st Century, pp. 227–248. Wiley, Chichester (2004)

  20. Frittelli, M., Scandolo, G.: Risk measures and capital requirements for processes. Math Finance 16(4), 589–612 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Givan, R., Leach, S., Dean, T.: Bounded-parameter Markov decision processes. Artif. Intell. 122(1–2), 71–109 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Harrison, J.M., Kreps, D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20(3), 381–408 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  23. Howard, R., Matheson, J.: Risk-sensitive Markov decision processes. Manag.Sci. 18(7), 356–369 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  24. Iyengar, G.N.: Robust dynamic programming. Math. Oper. Res. 30(2), 257–280 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jaquette, S.C.: A utility criterion for Markov decision processes. Manag. Sci. 23(1), 43–49 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mannor, S., Simester, D., Sun, P., Tsitsiklis, J.N.: Bias and variance approximation in value function estimates. Manag. Sci. 53(2), 308–322 (2007)

    Article  MATH  Google Scholar 

  27. Nilim, A., El Ghaoui, L.: Robustness in Markov decision problems with uncertain transition matrices. In: Advances in Neural Information Processing Systems, pp. 839–846 (2004)

  28. Nocedal, J., Wright, S.: Numerical Optimization Operations Research and Financial Engineering. Springer, New York (2006)

    Google Scholar 

  29. Pflug, G., Römisch, W.: Modeling, Measuring and Managing Risk. World Scientific, New York (2007)

    Book  MATH  Google Scholar 

  30. Pichler, A., Shapiro, A.: Risk averse stochastic programming: time consistency and optimal stopping, Preprint, arXiv:1808.10807 (2018)

  31. Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, Hoboken (2014)

    MATH  Google Scholar 

  32. Rockafellar, R., Uryasev, S., Zabarankin, M.: Generalized deviations in risk analysis. Finance Stoch. 10(1), 51–74 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Roorda, B., Schumacher, J.M., Engwerda, J.: Coherent acceptability measures in multi-period models. Math. Finance 15(4), 589–612 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ruszczyński, A.: Risk-averse dynamic programming for Markov decision processes. Math. Program. 125(2), 235–261 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Satia, J.K., Lave Jr., R.E.: Markov decision processes with uncertain transition probabilities. Oper. Res. 21(3), 728–740 (1973)

    Article  MATH  Google Scholar 

  36. Shapiro, A.: Worst-case distribution analysis of stochastic programs. Math. Program. 107(1–2), 91–96 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Shapiro, A.: Distributionally robust stochastic programming. SIAM J. Optim. 27(4), 2258–2275 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Staum, J.: Fundamental theorems of asset pricing for good deal bounds. Math. Finance 14(2), 141–161 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  39. Tamar, A., Mannor, S., Xu, H.: Scaling up robust MDPs using function approximation. In: International Conference on Machine Learning, pp. 181–189 (2014)

  40. Tseng, P.: Solving H-horizon, stationary Markov decision problems in time proportional to log(H). Oper. Res. Lett. 9(5), 287–297 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  41. West, D.: Updating mean and variance estimates: an improved method. Commun. ACM 22(9), 532–535 (1979)

    Article  MATH  Google Scholar 

  42. White III, C.C., Eldeib, H.K.: Markov decision processes with imprecise transition probabilities. Oper. Res. 42(4), 739–749 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wiesemann, W., Kuhn, D., Rustem, B.: Robust Markov decision processes. Math. Oper. Res. 38(1), 153–183 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wu, D., Koutsoukos, X.: Reachability analysis of uncertain systems using bounded-parameter Markov decision processes. Artif. Intell. 172(8–9), 945–954 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. Xu, H., Mannor, S.: Distributionally robust Markov decision processes. In: Advances in Neural Information Processing Systems, pp. 2505–2513 (2010)

  46. Yu, P., Xu, H.: Distributionally robust counterpart in Markov decision processes. IEEE Trans. Autom. Control 61(9), 2538–2543 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Industrial and Systems Engineering, Lehigh University, 200 W Packer Ave, Bethlehem, PA, 18015, USA

    Shu Tu & Boris Defourny

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  1. Shu Tu
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  2. Boris Defourny
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Correspondence to Boris Defourny.

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Tu, S., Defourny, B. An active-set strategy to solve Markov decision processes with good-deal risk measure. Optim Lett 13, 1239–1257 (2019). https://doi.org/10.1007/s11590-019-01413-0

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