Dynamic risk measures via backward doubly stochastic Volterra integral equations with jumps

References

• Acciaio, B., and I. Penner, 2011. Dynamic risk measures. In Advanced mathematical methods for finance, ed. G. Di Nunno and B. Øksendal, 1–34. Heidelberg: Springer.
   Google Scholar
• Acciaio, B., H. Föllmer, and I. Penner. 2012. Risk assessment for uncertain cash flows: model ambiguity, discounting ambiguity, and the role of bubbles. Finance and Stochastics 16:669–709. doi: 10.1007/s00780-012-0176-1.
   Web of Science ®Google Scholar
• Agram, N. 2019. Dynamic risk measure for BSVIE with jumps and semimartingale issues. Stochastic Analysis and Applications 37 (3):361–76. doi: 10.1080/07362994.2019.1569531.
   Web of Science ®Google Scholar
• Artzner, P., F. Dellbaen, J. M. Eber, and D. Heath. 1997. Thinking coherently. Risk 10:68–71.
   Google Scholar
• Artzner, P., F. Dellbaen, J. M. Eber, and D. Heath. 1999. Coherent measures of risk. Mathematical Finance 9 (3):203–28. doi: 10.1111/1467-9965.00068.
   Web of Science ®Google Scholar
• Barrieu, P., and N. El Karoui, 2009. Pricing, hedging, and designing derivatives with risk measures. In Indifference pricing: Theory and applications, ed. R. Carmona, 77–144. Princeton: Princeton University Press.
   Google Scholar
• Calvia, A., and E. Rosazza Gianin. 2020. Risk measures and progressive enlargement of filtration: a bsde approach. SIAM Journal on Financial Mathematics 11 (3):815–48. doi: 10.1137/19M1259134.
   Web of Science ®Google Scholar
• Cheridito, P., F. Delbaen, and K. Kupper. 2006. Dynamic monetary risk measures for bounded discrete-time processes. Electronic Journal of Probability 11 (3):57–106. doi: 10.1214/EJP.v11-302.
   Google Scholar
• Detlefsen, K., and G. Scandolo. 2005. Conditional and dynamic convex risk measures. Finance and Stochastics 9:539–61. doi: 10.1007/s00780-005-0159-6.
   Web of Science ®Google Scholar
• Fan, X. 2009. Comparison theorem of backward doubly stochastic differential equations driven by Lévy processes and application (in Chinese). Acta Mathematical Applicatae Sinica 32 (4):705–16.
   Google Scholar
• Fan, X., and J. Xu. 2015. Existence, uniqueness and comparison theorem of solutions for BDSDEs with jumps (in Chinese). Acta Mathematical Applicatae Sinica 38 (2):273–84.
   Google Scholar
• Föllmer, H, and A. Schied. 2002. Convex measures of risk and trading constrains. Finance and Stochastics 6:429–47. doi: 10.1007/s007800200072.
   Web of Science ®Google Scholar
• Föllmer, H., and A. Schied, 2016. Stochastic finance, an introduction in discrete time, 4th revised and extended ed. Berlin, New York: Welter de Gruyter.
   Google Scholar
• Frittelli, M., and E. Rosazza Gianin. 2002. Putting order in risk measures. Journal of Banking & Finance 26:1473–86. doi: 10.1016/S0378-4266(02)00270-4.
   Web of Science ®Google Scholar
• Ji, R., X. Shi, S. Wang, and J. Zhou. 2019. Dynamic risk measures for processes via backward stochastic differential equations. Insurance: Mathematics and Economics 86:43–50. doi: 10.1016/j.insmatheco.2019.02.005.
   Web of Science ®Google Scholar
• Jiang, L. 2008. Convexity, transalation invariance and subadditivity for g-expectations and related risk measures. Annals of Applied Probability 18 (1):245–58.
   Web of Science ®Google Scholar
• Øksendal, B., and A. Sulem. 2015. Risk minimization in financial markets modeled by Itô-Lévy processes. Afrika Matematika 26:939–79. doi: 10.1007/s13370-014-0248-9.
   Web of Science ®Google Scholar
• Pardoux, E., and S. Peng, 1994. Backward doubly stochastic differential equations and systems of quasilinear parabolic SPDE. Probability Theory and Related Fields, 98:209–27. doi: 10.1007/BF01192514.
   Web of Science ®Google Scholar
• Peng, S. 1997. BSDE and the related g-expectation. In Pitman Res. Notes Math. Ser., vol. 364, 141–59. Harlow: Longman.
   Google Scholar
• Penner, I., and A. Réveillac. 2015. Risk measures for processes and BSDEs. Finance and Stochastics 19 (1):23–66. doi: 10.1007/s00780-014-0243-x.
   Web of Science ®Google Scholar
• Quenez, M. C., and A. Sulem, 2013. BSDEs with jumps, optimization and applications to dynamic risk measures. Stochastic Processes and Their Applications 123:3328–57. doi: 10.1016/j.spa.2013.02.016.
   Web of Science ®Google Scholar
• Quenez, M. C., and A. Sulem, 2014. Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps. Stochastic Processes and Their Applications 124:3031–54. doi: 10.1016/j.spa.2014.04.007.
   Web of Science ®Google Scholar
• Ren, Y. 2010. On solutions of backward stochastic Volterra integral equations with jumps in Hilbert spaces. Journal of Optimization Theory and Applications 144:319–33. doi: 10.1007/s10957-009-9596-2.
   Web of Science ®Google Scholar
• Riedel, F. 2004. Dynamic coherent risk measures. Stochastic Processes and Their Applications 112: 185–200. doi: 10.1016/j.spa.2004.03.004.
   Web of Science ®Google Scholar
• Rosazza Gianin, E. 2006. Risk measures via g-expectations. Insurance: Mathematics and Economics 39 (1):19–34.
   Web of Science ®Google Scholar
• Shi, Y., J. Wen, and J. Xiong. 2020. Backward doubly stochastic Volterra integral equations and their applications. Journal of Differential Equations 269:6492–528. doi: 10.1016/j.jde.2020.05.006.
   Web of Science ®Google Scholar
• Sow, A. B., and Y. Sagna. 2018. BDSDE with Poisson jumps under stochastic Lipschitz and linear growth conditions. Stochastics and Dynamics 18 (5):1850039. doi: 10.1142/S0219493718500399.
   Web of Science ®Google Scholar
• Sun, X., and Y. Lu. 2008. The property for solutions of the multi-dimensional backward doubly stochastic differential equations with jumps (in Chinese). Chinese Journal of Applied Probability and Statistics 24 (1):73–82.
   Google Scholar
• Wang, T., and J. Yong. 2015. Comparison theorems for some backward stochastic Volterra integral equations. Stochastic Processes and Their Applications 125:1756–98. doi: 10.1016/j.spa.2014.11.013.
   Web of Science ®Google Scholar
• Wang, Z., and X. Zhang. 2007. Non-Lipschitz backward stochastic Volterra type equations with jumps. Stochastics and Dynamics 7 (4):479–96. doi: 10.1142/S0219493707002128.
   Web of Science ®Google Scholar
• Wen, J., and Y. Shi. 2020. Symmetrical martingale solutions of backward doubly stochastic Volterra integral equations. Computers & Mathematics with Applications 79 (5):1435–46. doi: 10.1016/j.camwa.2019.09.006.
   Web of Science ®Google Scholar
• Xu, Y. 2016. Multidimensional dynamic risk measure via conditional g-expectation. Mathematical Finance 26 (3):638–73. doi: 10.1111/mafi.12062.
   Web of Science ®Google Scholar
• Yong, J. 2006. Backward stochastic Volterra integral equations and some related problems. Stochastic Processes and Their Applications 116:779–95. doi: 10.1016/j.spa.2006.01.005.
   Web of Science ®Google Scholar
• Yong, J. 2007. Continuous-time dynamic risk measures by backward stochastic Volterra integral equations. Applicable Analysis 86 (11):1429–42. doi: 10.1080/00036810701697328.
   Web of Science ®Google Scholar
• Yong, J. 2008. Well-posedness and regularity of backward stochastic Volterra integral equations. Probability Theory and Related Fields 142:21–77. doi: 10.1007/s00440-007-0098-6.
   Web of Science ®Google Scholar
• Zhu, Q., and Y. Shi. 2012. Backward doubly stochastic differential equations with jumps and stochastic partial differential-integral equations. Chinese Annals of Mathematics, Series B 33 (1):127–42. doi: 10.1007/s11401-011-0686-8.
   Web of Science ®Google Scholar