In this paper, we study backward stochastic Volterra integral equations introduced in Lin [Stochastic Anal. Appl. 20 (2002) 165–183] and Yong [Stochastic Process. Appl. 116 (2006) 779–795] and extend the existence, uniqueness or comparison results for general filtration as in Papapantoleon et al. [Electron. J. Probab. 23 (2018) EJP240] (not only Brownian-Poisson setting). We also consider $$-data and explore the time regularity of the solution in the Itô setting, which is also new in this jump setting.
Keywords: Backward Volterra integral equation, general filtration, $$p-solution, jumps, time regularity
@article{PS_2021__25_1_133_0,
author = {Popier, Alexandre},
title = {Backward stochastic {Volterra} integral equations with jumps in a general filtration},
journal = {ESAIM: Probability and Statistics},
pages = {133--203},
year = {2021},
publisher = {EDP-Sciences},
volume = {25},
doi = {10.1051/ps/2021006},
mrnumber = {4234131},
zbl = {1469.45018},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2021006/}
}
TY - JOUR AU - Popier, Alexandre TI - Backward stochastic Volterra integral equations with jumps in a general filtration JO - ESAIM: Probability and Statistics PY - 2021 SP - 133 EP - 203 VL - 25 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ps/2021006/ DO - 10.1051/ps/2021006 LA - en ID - PS_2021__25_1_133_0 ER -
%0 Journal Article %A Popier, Alexandre %T Backward stochastic Volterra integral equations with jumps in a general filtration %J ESAIM: Probability and Statistics %D 2021 %P 133-203 %V 25 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ps/2021006/ %R 10.1051/ps/2021006 %G en %F PS_2021__25_1_133_0
Popier, Alexandre. Backward stochastic Volterra integral equations with jumps in a general filtration. ESAIM: Probability and Statistics, Tome 25 (2021), pp. 133-203. doi: 10.1051/ps/2021006
[1] and , High-Frequency Financial Econometrics. Princeton University Press, 1 edition (2014). | Zbl
[2] and , Semimartingale: Itô or not? Stochastic Process. Appl. 128 (2018) 233–254. | MR | Zbl | DOI
[3] and , Backward stochastic Volterra integral equations in Hilbert spaces. In Differential & difference equations and applications. Hindawi Publ. Corp., New York (2006) 57–66. | MR | Zbl
[4] , Bounded solutions to backward SDE’s with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16 (2006) 2027–2054. | MR | Zbl | DOI
[5] and , Volterra equations with Itô integrals. I. J. Integral Equ. 2 (1980) 187–245. | MR | Zbl
[6] and , Volterra equations with Itô integrals. II. J. Integr. Equ. 2 (1980) 319–337. | MR | Zbl
[7] and , Dirichlet forms methods for Poisson point measures and Lévy processes. With emphasis on the creation-annihilation techniques. Vol. 76 of Probability Theory and Stochastic Modelling. Springer, Cham (2015). | MR | Zbl | DOI
[8] , , , and , solutions of backward stochastic differential equations. Stochastic Process. Appl. 108 (2003) 109–129. | MR | Zbl | DOI
[9] , and , Backward stochastic differential equation driven by a marked point process: an elementary approach with an application to optimal control. Ann. Appl. Probab. 26 (2016) 1743–1773. | MR | Zbl | DOI
[10] and , Probabilités et potentiel. Théorie des martingales. Chapitres V à VIII. Hermann, Paris (1980). | MR
[11] , Backward stochastic differential equations with jumps and their actuarial and financial applications. BSDEs with jumps. European Actuarial Academy (EAA) Series. Springer, London (2013). | MR | Zbl
[12] , and , Malliavin calculus for Lévy processes with applications to finance. Universitext. Springer-Verlag, Berlin (2009). | MR | Zbl
[13] and , On a class of backward stochastic Volterra integral equations. Appl. Math. Lett. 26 (2013) 1192–1197. | MR | Zbl | DOI
[14] and , Backward stochastic Volterra integral equations with additive perturbations. Appl. Math. Comput. 265 (2015) 903–910. | MR | Zbl
[15] and , A general result of existence and uniqueness of backward stochastic differential equations. In Backward stochastic differential equations (Paris, 1995–1996). Volume 364 of Pitman Res. Notes Math. Ser. Longman, Harlow (1997) 27–36. | MR | Zbl
[16] , and , Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1–71. | MR | Zbl | DOI
[17] and , Linear Volterra backward stochastic integral equations. Stochastic Process. Appl. 129 (2019) 626–633. | MR | Zbl | DOI
[18] , Calcul stochastique et problèmes de martingales. Vol. 714 of Lecture Notes in Mathematics. Springer, Berlin (1979). | MR | Zbl
[19] and , Limit theorems for stochastic processes. Vol. 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition (2003). | MR | Zbl
[20] , Existence of solutions of stochastic Volterra integral equations. In Theory of random processes, No. 11. “ Naukova Dumka”, Kiev (1983) 51–57. | MR | Zbl
[21] , Existence of solutions of stochastic integral equations of Itô-Volterra type with locally integrable and continuous trajectories. In Theory of random processes, No. 12. “ Naukova Dumka”, Kiev (1984) 32–40. | MR | Zbl
[22] , and , Interpolation of linear operators. Translated from the Russian by J. Szűcs. Vol. 54 of Translations of Mathematical Monographs. American Mathematical Society, Providence, R.I. (1982). | MR | Zbl
[23] and , Bsdes with monotone generator driven by Brownian and Poisson noises in a general filtration. Stochastics 88 (2016) 491–539. | MR | Zbl | DOI
[24] and , -solution for BSDEs with jumps in the case : corrections to the paper ‘BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration. Stochastics 89 (2017) 1201–1227. | MR | Zbl | DOI
[25] , and , Présentation unifiée de certaines inégalités de la théorie des martingales. With an appendix by Lenglart. In Seminar on Probability, XIV (Paris, 1978/1979) (French). Vol. 784 of Lecture Notes in Math. Springer, Berlin (1980) 26–52. | MR | Numdam | Zbl
[26] , Adapted solution of a backward stochastic nonlinear Volterra integral equation. Stochastic Anal. Appl. 20 (2002) 165–183. | MR | Zbl | DOI
[27] and , Controlled singular volterra integral equations and pontryagin maximum principle. SIAM J. Control Optim. 58 (2020) 136–164. | MR | Zbl | DOI
[28] , Backward stochastic Volterra integral equations associated with a Levy process and applications. Preprint (2011). | arXiv
[29] and , On maximal inequalities for purely discontinuous martingales in infinite dimensions. In Séminaire de Probabilités XLVI. Vol. 2123 of Lecture Notes in Math. Springer, Cham (2014) 293–315. | MR | Zbl | DOI
[30] and , Path-dependent backward stochastic Volterra integral equations with jumps, differentiability and duality principle. Probab. Uncertain. Quant. Risk 3 (2018) 4. | MR | Zbl | DOI
[31] , and , Existence and uniqueness results for BSDE with jumps: the whole nine yards. Electr. J. Probab. 23 (2018) EJP240. | MR | Zbl
[32] and , Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990) 55–61. | MR | Zbl | DOI
[33] and , Stochastic Volterra equations with anticipating coefficients. Ann. Probab. 18 (1990) 1635–1655. | MR | Zbl | DOI
[34] and , Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Vol. 69 of Stochastic Modelling and Applied Probability. Springer-Verlag (2014). | MR | Zbl
[35] , Volterra equations driven by semimartingales. Ann. Probab. 13 (1985) 519–530. | MR | Zbl
[36] , Stochastic integration and differential equations. Vol. 21 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, second edition (2004). | Zbl | MR
[37] , On solutions of backward stochastic Volterra integral equations with jumps in Hilbert spaces. J. Optim. Theory Appl. 144 (2010) 319–333. | MR | Zbl | DOI
[38] and , Continuous martingales and Brownian motion. Vol. 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition (1999). | MR | Zbl | DOI
[39] , Theory of stochastic differential equations with jumps and applications. Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York (2005). | MR | Zbl
[40] , and , Recursive Utility Processes, Dynamic Risk Measures and Quadratic Backward Stochastic Volterra Integral Equations. Preprint (2018). | arXiv | Zbl
[41] , solutions of backward stochastic Volterra integral equations. Acta Math. Sin. (Engl. Ser.) 28 (2012) 1875–1882. | MR | Zbl | DOI
[42] and , Comparison theorems for some backward stochastic Volterra integral equations. Stochastic Process. Appl. 125 (2015) 1756–1798. | MR | Zbl | DOI
[43] and , Backward stochastic Volterra integral equations—representation of adapted solutions. Stochastic Process. Appl. 129 (2019) 4926–4964. | MR | Zbl | DOI
[44] and , Non-Lipschitz backward stochastic Volterra type equations with jumps. Stoch. Dyn. 7 (2007) 479–496. | MR | Zbl | DOI
[45] , Backward stochastic Volterra integral equations and some related problems. Stochastic Process. Appl. 116 (2006) 779–795. | MR | Zbl | DOI
[46] , Well-posedness and regularity of backward stochastic Volterra integral equations. Probab. Theory Related Fields 142 (2008) 21–77. | MR | Zbl | DOI
[47] , Backward stochastic Volterra integral equations—a brief survey. Appl. Math. J. Chinese Univ. Ser. B 28 (2013) 383–394. | MR | Zbl | DOI
Cité par Sources :
