Các thước đo rủi ro theo tập giá trị như các bao hàm và phương trình ngẫu nhiên lùi

Akahori, J., Amaba, T., Okuma, K.: A discrete-time Clark–Ocone formula and its application to an error analysis. J. Theor. Probab. 30, 932–960 (2017) Ararat, Ç., Hamel, A.H., Rudloff, B.: Set-valued shortfall and divergence risk measures. Int. J. Theor. Appl. Finance 20, 1750026 (2017) Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999) Barrieu, P., El Karoui, N.: Optimal derivative design under dynamic risk measures. In: Ying, G., Zhang, Q. (eds.) Mathematics of Finance, pp. 13–26. Am. Math. Soc., Providence (2004) Barrieu, P., El Karoui, N.: Pricing, hedging and optimally designing derivatives via minimization of risk measures. In: Carmona, R. (ed.) Indifference Pricing: Theory and Applications, pp. 77–146. Princeton University Press, Princeton (2009) Ben Tahar, I., Lépinette, E.: Vector-valued coherent risk measure processes. Int. J. Theor. Appl. Finance 17, 1450011–1–28 (2014) Biagini, F., Fouque, J.P., Frittelli, M., Meyer-Brandis, T.: A unified approach to systemic risk measures via acceptance sets. Math. Finance 9, 329–367 (2019) Bion-Nadal, J.: Conditional risk measures and robust representation of convex risk measures. Ecole Polytechnique CMAP preprint no. 557 (2004). Available online at http://www.cmap.polytechnique.fr/preprint/repository/557.pdf Cascos, I., Molchanov, I.: Multivariate risk measures: a constructive approach based on selections. Math. Finance 26, 867–900 (2016) Chen, Y., Hu, Y.: Time consistency for set-valued dynamic risk measures for bounded discrete-time processes. Math. Financ. Econ. 12, 305–333 (2018) Cheridito, P., Delbaen, F., Kupper, M.: Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11, 57–106 (2006) Cheridito, P., Horst, U., Kupper, M., Pirvu, T.A.: Equilibrium pricing in incomplete markets under translation invariant preferences. Math. Oper. Res. 41, 174–195 (2016) Crespi, G.P., Hamel, A.H., Schrage, C.: A Minty variational principle for set optimization. J. Math. Anal. Appl. 423, 770–796 (2015) Detlefsen, K., Scandolo, G.: Conditional and dynamic convex risk measures. Finance Stoch. 9, 539–561 (2005) Eisenberg, L., Noe, T.H.: Systemic risk in financial systems. Manag. Sci. 47, 236–249 (2001) Feinstein, Z., Rudloff, B.: Time consistency of dynamic risk measures in markets with transaction costs. Quant. Finance 13, 1473–1489 (2013) Feinstein, Z., Rudloff, B.: Multi-portfolio time consistency for set-valued convex and coherent risk measures. Finance Stoch. 19, 67–107 (2015) Feinstein, Z., Rudloff, B.: A comparison of techniques for dynamic multivariate risk measures. In: Hamel, A.H., et al. (eds.) Set Optimization and Applications – the State of the Art. From Set Relations to Set-valued Risk Measures, pp. 3–41. Springer, Berlin (2015) Feinstein, Z., Rudloff, B.: A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle. J. Glob. Optim. 68, 47–69 (2017) Feinstein, Z., Rudloff, B.: A supermartingale relation for multivariate risk measures. Quant. Finance 18, 1971–1990 (2018) Feinstein, Z., Rudloff, B.: Scalar multivariate risk measures with a single eligible asset. Working paper (2019). Available online at http://arxiv.org/abs/1807.10694 Feinstein, Z., Rudloff, B.: Time consistency for scalar multivariate risk measures. Working paper (2020). Available online at http://arxiv.org/abs/1810.04978 Feinstein, Z., Rudloff, B., Weber, S.: Measures of systemic risk. SIAM J. Financ. Math. 8, 672–708 (2017) Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stoch. 6, 429–447 (2002) Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. de Gruyter, Berlin (2011) Grépat, J., Kabanov, Yu.: On a multi-asset version of the Kusuoka limit theorem of option superreplication under transaction costs. Finance Stoch. 25, 167–187 (2021) Hamel, A.H., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Math. Financ. Econ. 5, 1–28 (2011) Hamel, A.H., Schrage, C.: Directional derivatives and subdifferential of set-valued convex functions. Pac. J. Optim. 10, 667–689 (2014) Hamel, A.H., Wang, S.Q.: A set optimization approach to utility maximization under transaction costs. Decis. Econ. Finance 40, 257–275 (2017) He, H.: Convergence from discrete to continuous time contingent claim prices. Rev. Financ. Stud. 3, 523–546 (1990) Jacod, J., Méléard, S., Protter, P.: Explicit form and robustness of martingale representations. Ann. Probab. 28, 1747–1780 (2000) Jiang, L.: Convexity, translation invariance and subadditivity for \(g\)-expectations and related risk measures. Ann. Appl. Probab. 18, 245–258 (2008) Jouini, E., Meddeb, M., Touzi, N.: Vector-valued coherent risk measures. Finance Stoch. 8, 531–552 (2004) Karnam, C., Ma, J., Zhang, J.: Dynamic approaches for some time-inconsistent optimization problems. Ann. Appl. Probab. 27, 3435–3477 (2017) Kisielewicz, M.: Backward stochastic differential inclusions. Dyn. Syst. Appl. 16, 121–140 (2007) Kisielewicz, M.: Weak compactness of weak solutions to backward stochastic differential inclusions. Dyn. Syst. Appl. 17, 351–370 (2008) Kisielewicz, M.: Stochastic Differential Inclusions and Applications. Springer, New York (2013) Kováčová, G., Rudloff, B.: Time consistency of the mean-risk problem. Oper. Res., forthcoming (2020). Available online at https://arxiv.org/abs/1806.10981 Lépinette, E., Molchanov, I.: Conditional cores and conditional convex hulls of random sets. J. Math. Anal. Appl. 478, 368–392 (2019) Lépinette, E., Molchanov, I.: Risk arbitrage and hedging to acceptability under transaction costs. Finance Stoch. 25, 101–132 (2021) Löhne, A., Rudloff, B.: An algorithm for calculating the set of superhedging portfolios in markets with transaction costs. Int. J. Theor. Appl. Finance 17, 1450012 (2014) Molchanov, I.: Theory of Random Sets, 2nd edn. Springer, London (2017) Molchanov, I., Mühlemann, A.: Nonlinear expectations of random sets. Finance Stoch. 25, 5–41 (2021) Peng, S.: Dynamical evaluations. C. R. Acad. Sci. Paris, Ser. I 339, 585–589 (2004) Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2005), version 2.1 Riedel, F.: Dynamic coherent risk measures. Stoch. Process. Appl. 112, 185–200 (2004) Rosazza Gianin, E.: Risk measures via \(g\)-expectations. Insur. Math. Econ. 39, 19–34 (2006) Ruszczyński, A., Shapiro, A.: Conditional risk mappings. Math. Oper. Res. 31, 544–561 (2006) Stadje, M.: Extending dynamic convex risk measures from discrete time to continuous time: a convergence approach. Insur. Math. Econ. 47, 391–404 (2010) Xu, Y.: Multidimensional dynamic risk measure via conditional \(g\)-expectation. Math. Finance 26, 638–673 (2016)