Operations Research Transactions >
2025 , Vol. 29 >Issue 3: 135 - 159
DOI: https://doi.org/10.15960/j.cnki.issn.1007-6093.2025.03.007
Some studies on stochastic optimization based quantitative risk management
Received date: 2025-03-21
Online published: 2025-09-09
Copyright
Risk management often plays an important role in decision making under uncertainty. In quantitative risk management, assessing and optimizing risk metrics requires efficient computing techniques and reliable theoretical guarantees. In this paper, we introduce several topics on quantitative risk management and review some of the recent studies and advancements on the topics. We consider several risk metrics and study decision models that involve the metrics, with a main focus on the related computing techniques and theoretical properties. We show that stochastic optimization, as a powerful tool, can be leveraged to effectively address these problems.
Zhaolin HU . Some studies on stochastic optimization based quantitative risk management[J]. Operations Research Transactions, 2025 , 29(3) : 135 -159 . DOI: 10.15960/j.cnki.issn.1007-6093.2025.03.007
| 1 | GlassermanP.Monte Carlo Methods in Financial Engineering[M].New York:Springer,2004. |
| 2 | RuszczyńskiA,ShapiroA.Optimization of convex risk functions[J].Mathematics of Operations Research,2006,31(3):433-452. |
| 3 | ShapiroA,DentchevaD,RuszczyńskiA.Lectures on Stochastic Programming: Modeling and Theory[M].Philadelphia:SIAM,2014. |
| 4 | JorionP.Value at Risk[M].New York:McGraw-Hill,2006. |
| 5 | RockafellarR T,UryasevS.Optimization of conditional value-at-risk[J].The Journal of Risk,2000,2(3):21-41. |
| 6 | HongL J,HuZ,LiuG.Monte Carlo methods for value-at-risk and conditional value-at-risk: A review[J].ACM Transactions on Modeling and Computer Simulation,2014,24(4):Article 22. |
| 7 | ArtznerP,DelbaenF,EberJ M,et al.Coherent measures of risk[J].Mathematical Finance,1999,9(3):203-228. |
| 8 | F?llmerH,SchiedA.Convex measures of risk and trading constraints[J].Finance and Stochastics,2002,6,429-447. |
| 9 | FrittelliM,GianinE R.Putting order in risk measures[J].Journal of Banking and Finance,2002,26(7):1473-1486. |
| 10 | CharnesA,CooperW W,SymondsG H.Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil[J].Management Science,1958,4(3):235-263. |
| 11 | Ben-TalA,TeboulleM.Expected utility, penalty functions, and duality in stochastic nonlinear programming[J].Management Science,1986,32(11):1445-1466. |
| 12 | Ben-Tal,A,TeboulleM.An old-new concept of convex risk measures: The optimized certainty equivalent[J].Mathematical Finance,2007,17(3):449-476. |
| 13 | Hamm A M, Salfeld T, Weber S. Stochastic root finding for optimized certainty equivalents[C]//Proceedings of the 2013 Winter Simulation Conference, 2013: 922-932. |
| 14 | RockafellarR T,UryasevS.The fundamental risk quadrangle in risk management, optimization and statistical estimation[J].Surveys in Operations Research and Management Science,2013,18(1):33-53. |
| 15 | RockafellarR T,RoysetJ O.Measures of residual risk with connections to regression, risk tracking, surrogate models, and ambiguity[J].SIAM Journal on Optimization,2015,25(2):1179-1208. |
| 16 | DunkelJ,WeberS.Stochastic root finding and efficient estimation of convex risk measures[J].Operations Research,2010,58(5):1505-1521. |
| 17 | TrindadeA A,UryasevS,ShapiroA,et al.Financial prediction with constrained tail risk[J].Journal of Banking and Finance,2007,31(11):3524-3538. |
| 18 | DurrettR.Probability: Theory and Examples[M].Cambridge:Cambridge University Press,2019. |
| 19 | HuberP J,RonchettiE M.Robust Statistics[M].New Jersey:John Wiley & Sons,2011. |
| 20 | HongL J,LiuG.Simulating sensitivities of conditional value-at-risk[J].Management Science,2009,55(2):281-293. |
| 21 | GlynnP W,FanL,FuM C,et al.Central limit theorems for estimated functions at estimated points[J].Operations Research,2020,68(5):1557-1563. |
| 22 | KrokhmalP A.Higher moment coherent risk measures[J].Quantitative Finance,2007,7(4):373-387. |
| 23 | AlexanderS,ColemanT F,LiY.Minimizing CVaR and VaR for a portfolio of derivatives[J].Journal of Banking and Finance,2006,30(2):583-605. |
| 24 | HuZ,ZhangD.Utility-based shortfall risk: Efficient computations via Monte Carlo[J].Naval Research Logistics,2018,65(5):378-392. |
| 25 | Hegde V, Menon A S, Prashanth L A, et al. Online estimation and optimization of utility-based shortfall risk[J/OL].[2025-08-05]. Mathematics of Operations Research. |
| 26 | LuedtkeJ,AhmedS.A sample approximation approach for optimization with probabilistic constraints[J].SIAM Journal on Optimization,2008,19(2):674-699. |
| 27 | PagnoncelliB K,AhmedS,ShapiroA.Sample average approximation method for chance constrained programming: Theory and applications[J].Journal of Optimization Theory and Applications,2009,142,399-416. |
| 28 | CalafioreG,CampiM C.Uncertain convex programs: Randomized solutions and confidence levels[J].Mathematical Programming,2005,102,25-46. |
| 29 | CalafioreG,CampiM C.The scenario approach to robust control design[J].IEEE Transactions on Automatic Control,2006,51(5):742-753. |
| 30 | De FariasD P,Van RoyB.On constraint sampling in the linear programming approach to approximate dynamic programming[J].Mathematics of Operations Research,2004,29(3):462-478. |
| 31 | Kü?ükyavuzS,JiangR.Chance-constrained optimization under limited distributional information: A review of reformulations based on sampling and distributional robustness[J].EURO Journal on Computational Optimization,2022,10,100030. |
| 32 | HenrionR,M?llerA.A gradient formula for linear chance constraints under Gaussian distribution[J].Mathematics of Operations Research,2012,37(3):475-488. |
| 33 | van AckooijW,HenrionR.Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distributions[J].SIAM Journal on Optimization,2014,24(4):1864-1889. |
| 34 | HongL J.Estimating quantile sensitivities[J].Operations Research,2009,57(1):118-130. |
| 35 | HongL J,JiangG.Gradient and hessian of joint probability function with applications on chance-constrained programs[J].Journal of the Operations Research Society of China,2017,5,431-455. |
| 36 | Feng G, Liu G. Conditional Monte Carlo: A change-of-variables approach[EB/OL].[2025-08-05]. |
| 37 | HongL J,YangY,ZhangL.Sequential convex approximations to joint chance constrained programs: A Monte Carlo approach[J].Operations Research,2011,59(3):617-630. |
| 38 | HuZ,HongL J,ZhangL.A smooth Monte Carlo approach to joint chance constrained program[J].IIE Transactions,2013,45(7):716-735. |
| 39 | ShanF,ZhangL,XiaoX.A smoothing function approach to joint chance-constrained programs[J].Journal of Optimization Theory and Applications,2014,163,181-199. |
| 40 | CuiY,LiuJ,PangJ S.Nonconvex and nonsmooth approaches for affine chance-constrained stochastic programs[J].Set-Valued and Variational Analysis,2022,30(3):1149-1211. |
| 41 | Pe?a-OrdieresA,LuedtkeJ,W?chterA.Solving chance-constrained problems via a smooth sample-based nonlinear approximation[J].SIAM Journal on Optimization,2020,30(3):2221-2250. |
| 42 | HuZ,SunW,ZhuS.Chance constrained programs with Gaussian mixture models[J].IISE Transactions,2022,54(12):1117-1130. |
| 43 | WeiJ,HuZ,LuoJ.Appointment scheduling optimization with chance constraints in a singleserver consultation system[J].Systems Engineering-Theory & Practice,2024,44(10):3400-3417. |
| 44 | WeiJ,HuZ,LuoJ,et al.Enhanced branch-and-bound algorithm for chance constrained programs with Gaussian mixture models[J].Annals of Operations Research,2024,338(2):1283-1315. |
| 45 | Pang X, Zhu S, Hu Z. Chance constrained program with quadratic randomness: A unified approach based on Gaussian mixture distribution[EB/OL].[2025-07-06]. arXiv:2303.00555v1. |
| 46 | GordyM B,JunejaS.Nested simulation in portfolio risk measurement[J].Management Science,2010,56(9):1658-1673. |
| 47 | BroadieM,DuY,MoallemiC C.Risk estimation via regression[J].Operations Research,2015,63(5):1077-1097. |
| 48 | HongL J,JunejaS,LiuG.Kernel smoothing for nested estimation with application to portfolio risk measurement[J].Operations Research,2017,65(3):657-673. |
| 49 | ZhangK,LiuG,WangS.Bootstrap-based budget allocation for nested simulation[J].Operations Research,2022,70(2):1128-1142. |
| 50 | WangW,WangY,ZhangX.Smooth nested simulation: Bridging cubic and square root convergence rates in high dimensions[J].Management Science,2024,70(2):9031-9057. |
| 51 | Liu G, Zhang K. A tutorial on nested simulation[C]//Proceedings of the 2024 Winter Simulation Conference, 2024: 1-15. |
| 52 | HuZ,HongL J.Robust simulation with likelihood-ratio constrained input uncertainty[J].INFORMS Journal on Computing,2022,34(4):2350-2367. |
| 53 | KuhnD,ShafieeS,WiesemannW.Distributionally robust optimization[J].Acta Numerica,2025,34,579-804. |
| 54 | ZhuS,FukushimaM.Worst-case conditional value-at-risk with application to robust portfolio management[J].Operations Research,2009,57(5):1155-1168. |
| 55 | GuoS,XuH.Distributionally robust shortfall risk optimization model and its approximation[J].Mathematical Programming,2019,174(1):473-498. |
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