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The perturbation method applied to a robust optimization problem with constraint

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Abstract

The present paper studies a kind of robust optimization problems with constraint. The problem is formulated through Backward Stochastic Differential Equations (BSDEs) with quadratic generators. A necessary condition is established for the optimal solution using a terminal perturbation method and properties of Bounded Mean Oscillation (BMO) martingales. The necessary condition is further proved to be sufficient for the existence of an optimal solution under an additional convexity assumption. Finally, the optimality condition is applied to discuss problems of partial hedging with ambiguity, fundraising under ambiguity and randomized testing problems for a quadratic g-expectation.

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References

  1. Barrieu, P., El Karoui, N.: Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs. Ann. Probab. 41, 1831–1863 (2013)

    Article  MathSciNet  Google Scholar 

  2. Bernard, C., Ji, S., Tian, W.: An optimal insurance design problem under Knightian uncertainty. Dec. Econ. Finan. 36(2), 99–124 (2013)

    Article  MathSciNet  Google Scholar 

  3. Bordigoni, G., Matoussi, A., Schweizer, M.: A stochastic control approach to a robust utility maximization problem. In: Stochastic Analysis and Applications (pp. 125–151). Springer, Berlin, Heidelberg (2007)

  4. Briand, P., Elie, R.: A simple constructive approach to quadratic BSDEs with or without delay. Stoch. Process. Appl. 123, 2921–2939 (2013)

    Article  MathSciNet  Google Scholar 

  5. Chen, Z., Epstein, L.: Ambiguity, risk, and asset returns in continuous time. Econometrica 70(4), 1403–1443 (2002)

  6. Chen, Z., Kulperger, R.: Minimax pricing and Choquet pricing. Insur.: Math. Econ. 38(3), 518–528 (2006)

    MathSciNet  Google Scholar 

  7. Chen, Z., Chen, T., Davison, M.: Choquet expectation and Peng’s g-expectation. Ann. Probab. 33(3), 1179–1199 (2005)

    Article  MathSciNet  Google Scholar 

  8. Cong, J., Tan, K.S., Weng, C.: Conditional value-at-risk-based optimal partial hedging. J. Risk 16(3), 49–83 (2014)

  9. Cong, J., Tan, K.S., Weng, C.: VaR-based optimal partial hedging. Astin Bull. 43(03), 271–299 (2013)

    Article  MathSciNet  Google Scholar 

  10. Cvitanic, J., Karatzas, I.: Generalized Neyman–Pearson lemma via convex duality. Bernoulli 7, 79–97 (2001)

    Article  MathSciNet  Google Scholar 

  11. Delbaen, F., Peng, S., Rosazza Gianin, E.: Representation of the penalty term of dynamic concave utilities. Finance Stoch. 14(3), 449–472 (2010)

    Article  MathSciNet  Google Scholar 

  12. El Karoui, N., Peng, S., Quenez, M.C.: A dynamic maximum principle for the optimization of recursive utilities under constraints. Ann. Appl. Probab. 664–693 (2001)

  13. Embrechts, P., Schied, A., Wang, R.: Robustness in the optimization of risk measures. Oper. Res. 70(1), 95–110 (2022)

    Article  MathSciNet  Google Scholar 

  14. Faidi, W., Matoussi, A., Mnif, M.: Maximization of recursive utilities. A dynamic maximum principle approach. SIAM J. Financ. Math. 2, 1014–1041 (2011)

    Article  MathSciNet  Google Scholar 

  15. Faidi, W., Mezghanni, H., Mnif, M.: Expected utility maximization problem under state constraints and model uncertainty. J. Optim. Theory Appl. 183(3), 1123–52 (2019)

    Article  MathSciNet  Google Scholar 

  16. Föllmer, H., Schied, A., Weber, S.: Robust preferences and robust portfolio choice. In: Ciarlet, P., Bensoussan, A., Zhang, Q. (eds.) Mathematical Modelling and Numerical Methods in Finance, vol. 15, pp. 29–88. Handbook of Numerical Analysis (2009)

  17. Föllmer, H., Leukert, P.: Quantile hedging. Financ. Stoch. 3(3), 251–273 (1999)

    Article  MathSciNet  Google Scholar 

  18. He, K., Hu, M., Chen, Z.: The relationship between risk measures and choquet expectations in the framework of g-expectations. Stat. Probab. Lett. 79(4), 508–512 (2009)

    Article  MathSciNet  Google Scholar 

  19. Ji, S.: Dual method for continuous-time Markowitz’s problems with nonlinear wealth equations. J. Math. Anal. Appl. 366(1), 90–100 (2010)

    Article  MathSciNet  Google Scholar 

  20. Ji, S., Peng, S.: Terminal perturbation method for the backward approach to continuous time mean-variance portfolio selection. Stoch. Process. Appl. 118(6), 952–967 (2008)

    Article  MathSciNet  Google Scholar 

  21. Ji, S., Zhou, X.Y.: A maximum principle for stochastic optimal control with terminal state constraints, and its applications. Commun. Inf. Syst. 6(4), 321–338 (2006)

    Article  MathSciNet  Google Scholar 

  22. Ji, S., Zhou, X.Y.: A generalized Neyman–Pearson lemma for g-probabilities. Probab. Theory Relat. Fields 148(3–4), 645–669 (2010)

    Article  MathSciNet  Google Scholar 

  23. Jiang, L.: Convexity, translation invariance and subadditivity for G-expectations and related risk measures. Ann. Appl. Probab. 18(1), 245–258 (2008)

  24. Kazamaki, N.: Continuous exponential martingales and BMO. In: Lecture Notes in Mathematics, vol. 1570. Springer, Berlin (1994)

  25. Kobylanski, M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28(2), 558–602 (2000)

    Article  MathSciNet  Google Scholar 

  26. Kulldorff, M.: Optimal control of favorable games with a time limit. SIAM J. Control. Optim. 31, 52–69 (1993)

    Article  MathSciNet  Google Scholar 

  27. Kupper, M., Schachermayer, W.: Representation results for law invariant time consistent functions. Math. Financ. Econ. 2(3), 189–210 (2009)

    Article  MathSciNet  Google Scholar 

  28. Ma, J., Yao, S.: On quadratic \(g\)-evaluation/expectations and related analysis. Stoch. Anal. Appl. 28(4), 711–734 (2010)

    Article  MathSciNet  Google Scholar 

  29. Melnikov, A., Smirnov, I.: Dynamic hedging of conditional value-at-risk. Insur.: Math. Econ. 51(1), 182–190 (2012)

    MathSciNet  Google Scholar 

  30. Peng, S.: Backward SDE and related g-expectation. Backward stochastic differential equations. Pitman Res. Notes Math. 364, 141–159 (1997)

  31. Peng, S.: Nonlinear, expectations, nonlinear evaluations, measures, risk. In: Stochastic Methods in Finance. Lecture Notes in Mathematics, vol. 2004. Springer, Berlin, Heidelberg (1856)

  32. Rosazza Gianin, E.: Risk measures via g-expectations. Insur.: Math. Econ. 39(1), 19–34 (2006)

    MathSciNet  Google Scholar 

  33. Rudloff, B.: Convex hedging in incomplete markets. Appl. Math. Financ. 14(5), 437–452 (2007)

    Article  MathSciNet  Google Scholar 

  34. Schied, A.: Risk measures and robust optimization problems. Stoch. Model. 22, 753–831 (2006)

    Article  MathSciNet  Google Scholar 

  35. Schroder, M., Skiadas, C.: Optimal lifetime consumption-portfolio strategies under trading constraints and generalized recursive preferences. Stoch. Process. Appl. 108(2), 155–202 (2003)

    Article  MathSciNet  Google Scholar 

  36. Sekine, J.: Dynamic minimization of worst conditional expectation of shortfall. Math. Financ. 14, 605–618 (2004)

    Article  MathSciNet  Google Scholar 

  37. Skiadas, C.: Robust control and recursive utility. Financ. Stoch. 7, 475–489 (2003)

    Article  MathSciNet  Google Scholar 

  38. Yong, J., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer (1999)

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Acknowledgements

We are highly grateful to Prof. Chengguo Weng for his helpful suggestions and comments. The authors moreover thank two anonymous referees for their helpful remarks and comments.

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

    Peng Luo

  2. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Alexander Schied

  3. School of Management, Shandong University, Jinan, 250100, China

    Xiaole Xue

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  1. Peng Luo
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  2. Alexander Schied
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  3. Xiaole Xue
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Correspondence to Xiaole Xue.

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The authors gratefully acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada through grant RGPIN-2017-04054. Peng Luo gratefully acknowledges the support from the National Natural Science Foundation of China (Grant No. 12101400). Xiaole Xue gratefully acknowledges the support from National Natural Science Foundation of China (No. 12001316), “The Fundamental Research Funds of Shandong University”.

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Luo, P., Schied, A. & Xue, X. The perturbation method applied to a robust optimization problem with constraint. Math Finan Econ (2024). https://doi.org/10.1007/s11579-024-00358-y

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