ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES*

doi:10.1007/s10473-023-0321-2

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (3): 1365-1381.doi: 10.1007/s10473-023-0321-2

ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES^*

Liangliang Mia¹, Yanhong Chen^2,†, Xiao Xiao³, Yijun Hu⁴

1. School of Mathematics and Information Technology, Jiangsu Second Normal University, Nanjing 210013, China;
2. College of Finance and Statistics, Hunan University, Changsha 410082, China;
3. School of Mathematics and Information Technology, Jiangsu Second Normal University, Nanjing 210013, China;
4. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

收稿日期:2020-09-17 修回日期:2022-01-18 出版日期:2023-06-25 发布日期:2023-06-06
通讯作者: ^† Yanhong Chen, E-mail: yhchen@hnu.edu.cn
作者简介:Liangliang Miao, E-mail: liangliangmiao@whu.edu.cn;Xiao Xiao, E-mail: primexxiao@163.com;Yijun Hu, E-mail: yjhu.math@whu.edu.cn
基金资助:
This paper was supported by the National Natural Science Foundation of China (11901184, 11771343) and the Natural Science Foundation of Hunan Province (2020JJ5025).

ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES^*

Liangliang Mia¹, Yanhong Chen^2,†, Xiao Xiao³, Yijun Hu⁴

1. School of Mathematics and Information Technology, Jiangsu Second Normal University, Nanjing 210013, China;
2. College of Finance and Statistics, Hunan University, Changsha 410082, China;
3. School of Mathematics and Information Technology, Jiangsu Second Normal University, Nanjing 210013, China;
4. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received:2020-09-17 Revised:2022-01-18 Online:2023-06-25 Published:2023-06-06
Contact: ^† Yanhong Chen, E-mail: yhchen@hnu.edu.cn
About author:Liangliang Miao, E-mail: liangliangmiao@whu.edu.cn;Xiao Xiao, E-mail: primexxiao@163.com;Yijun Hu, E-mail: yjhu.math@whu.edu.cn
Supported by:
This paper was supported by the National Natural Science Foundation of China (11901184, 11771343) and the Natural Science Foundation of Hunan Province (2020JJ5025).

摘要/Abstract

摘要： In this paper, we focus on anticipated backward stochastic Volterra integral equations (ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps. Motivated by this, a class of dynamic risk measures induced by ABSVIEs with jumps are discussed.

关键词: anticipated backward stochastic Volterra integral equations, comparison theorems, dynamic risk measures

Abstract: In this paper, we focus on anticipated backward stochastic Volterra integral equations (ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps. Motivated by this, a class of dynamic risk measures induced by ABSVIEs with jumps are discussed.

Key words: anticipated backward stochastic Volterra integral equations, comparison theorems, dynamic risk measures

Liangliang Mia, Yanhong Chen, Xiao Xiao, Yijun Hu. ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES^*[J]. 数学物理学报(英文版), 2023, 43(3): 1365-1381.

Liangliang Mia, Yanhong Chen, Xiao Xiao, Yijun Hu. ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES^*[J]. Acta mathematica scientia,Series B, 2023, 43(3): 1365-1381.

参考文献

[1] Agram N.Dynamic risk measure for BSVIE with jumps and semimartinggale issues. Stoch Anal Appl, 2019, 37(3): 361-376
[2] Artzner P, Delbaen F, Eber J M, Heath D.Coherent measures of risk. Math Finance, 1999, 9(3): 203-228
[3] Barles G, Buckdahn R, Pardoux E.Backward stochastic differential equations and integral-partial differ- ential equations. Stochastics Stochastics Rep, 1997, 60: 57-83
[4] Calvia A, Gianin E R.Risk measures and progressive enlargement of filtration: A BSDE approach. SIAM J Financial Math, 2020, 11: 815-848
[5] Delong L.Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications. New York: Springer, 2013
[6] El Karoui N, Peng S, Quenez M C.Backward stochastic differential equations in finance. Math Finance, 1997, 7(1): 1-71
[7] Füollmer H, Schied A.Convex measures of risk and trading constraints. Finance Stoch, 2002, 6: 429-447
[8] Füollmer H, Schied A.Stochastic Finance: An Introduction in Discrete Time. Berlin: De Gruyter, 2016
[9] Frittelli M, Gianin E R.Putting order in risk measures. J Banking Finance, 2002, 26: 1473-1486
[10] Gianin E R.Risk measures via g-expectations. Insur Math Econ, 2006, 39: 19-34
[11] Hamaguchi Y.Extended backward stochastic Volterra integral equations and their applications to time- inconsistent stochastic recursive control problems. Math Control Relat Fields, 2020, 11: 433-478
[12] Hamaguchi Y.Infinite horizon backward stochastic Volterra integral equations and discounted control problems. ESAIM Control Optim Calc Var, 2021, 27: 1-47
[13] Ji R, Shi X, Wang S, Zhou J.Dynamic risk measures for processes via backward stochastic differentical equations. Insur Math Econ, 2019, 86: 43-50
[14] Jiang L.Convexity, translation invariance and subadditivity for g-expectations and related risk measures. Ann Appl Probab, 2008, 18(1): 245-258
[15] Lin J.Adapted solutions of a backward stochastic nonlinear Volterra integral equation. Stoch Anal Appl, 2001, 20: 165-183
[16] Overbeck L, Rüoder J A L. Path-dependent backward stochastic Volterra integral equations with jumps, differentiability and duality principle. Probab Uncertain Quant Risk, 2018, 3(1): Art 4
[17] Pardoux E, Peng S.Adapted solution of a backward stochastic differential equation. Syst Control Lett, 1990, 14: 55-61
[18] Peng S.Backward SDE and related g-expectation//El Karoui N, Mazliak L. Backward Stochastic Dierential Equations. London: Longman, 1997, 364: 141-159
[19] Peng S, Yang Z.Anticipated backward stochastic differential equations. Ann Probab, 2009, 37: 877-902
[20] Penner I, Réveillac A.Risk measure for processes and BSDEs. Finance and Stochastics, 2015, 19: 23-66
[21] Popier A.Backward stochastic Volterra integral equations with jumps in a general filltration. ESAIM Probab Stat, 2021, 25: 133-203
[22] Quenez M C, Sulem A.BSDEs with jumps, optimization and applications to dynamic risk measures. Stochastic Process Appl, 2013, 123: 3328-3357
[23] Quenez M C, Sulem A.Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps. Stochastic Process Appl, 2014, 124: 3031-3054
[24] Ren Y.On solutions of backward stochastic Volterra integral equations with jumps in Hilbert spaces. J Optim Theory Appl, 2010, 144: 319-333
[25] Royer M.Backward stochastic differential equations with jumps and related non-linear expectations. Stochastic Process Appl, 2006, 116: 1358-1376
[26] Shi Y, Wang T.Solvability of general backward stochastic volterra integral equations. J Korean Math Soc, 2012, 49: 1301-1321
[27] Shi Y, Wen J, Xiong J.Optimal control problems of forward-backward stochastic Volterra integral equa- tions. Math Control Relat Fields, 2015, 5: 613-649
[28] Shi Y, Wen J, Xiong J.Backward doubly stochastic Volterra integral equations and applications to optimal control problems. J Differential Equations, 2020, 269: 6492-6528
[29] Situ R.Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering. New York: Springer, 2005
[30] Tang S, Li X.Necessary conditions for optimal control of stochastic systems with random jumps. SIAMJ Control Optim, 1994, 32: 1447-1475
[31] Wang H.Extended backward stochastic Volterra integral equations, quasilinear parabolic equations, and Feynman-Kac formula. Stoch Dyn, 2020, 21: 2150004
[32] Wang H, Sun J, Yong J.Recursive utility processes, dynamic risk measures and quadratic backward stochas- tic Volterra integral equations. Appl Math Optim, 2021, 84: 145-190
[33] Wang H, Yong J.Time-inconsistent stochastic optimal control problems and backward stochastic Volterra integral equations. ESAIM Control Optim Calc Var, 2021, 27: 1-40
[34] Wang T, Shi Y.A class of time inconsistent risk measures and backward stochastic Volterra integral equations. Risk Decis Anal, 2013, 4: 17-24
[35] Wang T, Yong J.Comparison theorems for some backward stochastic Volterra integral equations. Stoch Process Appl, 2015, 125: 1756-1798
[36] Wang T, Zhang H.Optimal control problems of forward-backward stochastic Volterra integral equations with closed control regions. SIAM J Control Optim, 2017, 55: 2574-2602
[37] Wang Z, Zhang X.Non-Lipschitz backward stochastic Volterra type equations with jumps. Stoch Dyn, 2007, 7: 479-496
[38] Wen J, Shi Y.Solvability of anticipated backward stochastic Volterra integral equations. Statist Probab Lett, 2020, 156: 108599
[39] Xu Y.Multidimensional dynamic risk measures via conditional g-expectation. Math Finance, 2016, 26: 638-673
[40] Yong J.Backward stochastic Volterra integral equations and some related problems. Stoch Proc Appl, 2006, 116: 779-795
[41] Yong J.Continuous-time dynamic risk measures by backward stochastic Volterra integral equations. Appl Anal, 2007, 86: 1429-1442
[42] Yong J.Well-posedness and regularity of backward stochastic Volterra integral equations. Probab Theory Relat Fields, 2008, 142: 21-77
[43] Yong J.Backward stochastic Volterra integral equations-a brief survey. Appl Math J Chin Univ, 2013, 28: 383-394

ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES^*

ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES^*

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