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Path-dependent backward stochastic Volterra integral equations with jumps, differentiability and duality principle
Institute of Mathematics, University of Gießen, Arndtsraße 2, 35392 Gießen, Germany |
References:
[1] |
Aman, A, N'Zi, M:Backward stochastic nonlinear Volterra integral equations with local Lipschitz drift. Probab. Math. Stat. 25, 105-127 (2005) Google Scholar |
[2] |
Ankirchner, S, Imkeller, P, Dos Reis, G:Classical and variational differentiability of BSDEs with quadratic growth. Electron. J. Probab. 12, 1418-1453 (2007) Google Scholar |
[3] |
Barles, G, Buckdahn, R, Pardoux, E:Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60, 57-83 (1997) Google Scholar |
[4] |
Becherer, D:Bounded solutions to bsdes with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16, 2027-2054 (2006) Google Scholar |
[5] |
Berger, M, Mizel, V:Volterra equations with itô integrals. J Integr. Equ. 2, 187-245 (1980) Google Scholar |
[6] |
Carmona, R:Indifference Pricing:Theory and Applications. Princeton University Press, Princeton (2008) Google Scholar |
[7] |
Cont, R, Fournié, DA:Change of variable formulas for non-anticipative functionals on path space. J. Funct. Anal. 259, 1043-1072 (2010) Google Scholar |
[8] |
Crépey, S:Financial Modeling, a backward stochastic differential equations perspective. Springer-Verlag Berlin Heidelberg (2013) Google Scholar |
[9] |
Delong, L:Backward stochastic differential equations with jumps and their actuarial and financial applications. Springer-Verlag London, London (2013) Google Scholar |
[10] |
Doléans-Dade, C:Quelques applications de la formule de changement de variables pour les semimartin-gales. Z Wahrscheinlichkeitstheorie verw Gebiete. 16, 181-194 (1970) Google Scholar |
[11] |
Duffie, D, Epstein, LG:Stochastic differential utility. Econometrica. 60(2), 353-394 (1992) Google Scholar |
[12] |
Dupire, B:Functional Itô calculus (2009). Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS. Available at SSRN:https://ssrn.com/abstract=1435551 Google Scholar |
[13] |
El Karoui, N, Peng, S, Quenez, M:Backward stochastic differential equations in finance. Math. Financ. 7, 1-71 (1997) Google Scholar |
[14] |
He, S, Wang, J, Yan, J:Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton (1992) Google Scholar |
[15] |
Jacod, J, Shiryaev, AN:Limit theorems for stochastic processes, 2nd edn. Springer-Verlag Berlin Heidelberg, Berlin (2003) Google Scholar |
[16] |
Kromer, E, Overbeck, L:Representation of BSDE-based dynamic risk measures and dynamic capital allocations. Int. J. Theor. Appl. Financ. 17, 1-16 (2014) Google Scholar |
[17] |
Kromer, E, Overbeck, L:Differentiability of BSVIEs and dynamic capital allocations. Int. J. Theor. Appl. Financ. 20, 1-26 (2017) Google Scholar |
[18] |
Kromer, E, Overbeck, L, Röder, J:Feynman Kac for functional jump diffusions with an application to credit value adjustment. Stat. Probab. Lett. 105, 120-129 (2015) Google Scholar |
[19] |
Kromer, E, Overbeck, L, Röder, J:Path-dependent BSDEs with jumps and their connection to PPIDEs. Stochast. Dyn. 17, 1-37 (2017) Google Scholar |
[20] |
Lazrak, A, Quenez, M:A generalized stochastic differential utility. Math. Oper. Res. 28, 154-180 (2003) Google Scholar |
[21] |
Levental, S, Schroder, M, Sinha, S:A simple proof of functional Itô's lemma for semimartingales with an application. Stat. Probab. Lett. 83, 2019-2026 (2013) Google Scholar |
[22] |
Lin, J:Adapted solution of a backward stochastic nonlinear Volterra integral equation. Stoch. Anal. Appl. 20, 165-183 (2002) Google Scholar |
[23] |
Lu, W:Backward stochastic Volterra integral equations associated with a Lévy process and applications (2016). Available at arXiv:https://arxiv.org/abs/1106.6129 Google Scholar |
[24] |
Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55-61 (1990) Google Scholar |
[25] |
Pardoux, E, Protter, P:Stochastic Volterra equations with anticipating coefficients. Ann. Probab. 18, 1635-1655 (1990) Google Scholar |
[26] |
Peng, S, Wang, F:BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. Sci. China Math. 59, 1-18 (2016) Google Scholar |
[27] |
Protter, P:Volterra equations driven by semimartingales. Ann. Probab. 13, 519-530 (1985) Google Scholar |
[28] |
Ren, Y:On solutions of backward stochastic volterra integral equations with jumps in hilbert spaces. J. Optim. Theory Appl. 144, 319-333 (2010) Google Scholar |
[29] |
Rong, S:On solutions of backward stochastic differential equations with jumps and applications. Stoch. Process. Appl. 66, 209-236 (1997) Google Scholar |
[30] |
Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control. Optim. 32, 1447-1475 (1994). https://doi.org/101137/S0363012992233858 Google Scholar |
[31] |
Wang, F:BSDEs with jumps and path-dependent parabolic integro-differential equations. Chin. Ann. Math. Ser. B. 36, 625-644 (2015) Google Scholar |
[32] |
Wang, T, Yong, J:Comparison theorems for some backward stochastic Volterra integral equations. Stoch. Process. Appl. 125, 1756-1798 (2015) Google Scholar |
[33] |
Wang, Z, Zhang, X:Non-Lipschitz backward stochastic Volterra type equations with jumps. Stochastics and Dynamics. 7(4), 479-496 (2007) Google Scholar |
[34] |
Yong, J:Backward stochastic Volterra integral equations and some related problems. Stoch. Process. Appl. 116, 779-795 (2006) Google Scholar |
[35] |
Yong, J:Continuous-time dynamic risk measures by backward stochastic Volterra integral equations. Appl Anal. 186, 1429-1442 (2007) Google Scholar |
[36] |
Yong, J:Well-posedness and regularity of backward stochastic volterra integral equations. Probab. Theory Relat. Fields. 142, 21-77 (2008) Google Scholar |
show all references
References:
[1] |
Aman, A, N'Zi, M:Backward stochastic nonlinear Volterra integral equations with local Lipschitz drift. Probab. Math. Stat. 25, 105-127 (2005) Google Scholar |
[2] |
Ankirchner, S, Imkeller, P, Dos Reis, G:Classical and variational differentiability of BSDEs with quadratic growth. Electron. J. Probab. 12, 1418-1453 (2007) Google Scholar |
[3] |
Barles, G, Buckdahn, R, Pardoux, E:Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60, 57-83 (1997) Google Scholar |
[4] |
Becherer, D:Bounded solutions to bsdes with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16, 2027-2054 (2006) Google Scholar |
[5] |
Berger, M, Mizel, V:Volterra equations with itô integrals. J Integr. Equ. 2, 187-245 (1980) Google Scholar |
[6] |
Carmona, R:Indifference Pricing:Theory and Applications. Princeton University Press, Princeton (2008) Google Scholar |
[7] |
Cont, R, Fournié, DA:Change of variable formulas for non-anticipative functionals on path space. J. Funct. Anal. 259, 1043-1072 (2010) Google Scholar |
[8] |
Crépey, S:Financial Modeling, a backward stochastic differential equations perspective. Springer-Verlag Berlin Heidelberg (2013) Google Scholar |
[9] |
Delong, L:Backward stochastic differential equations with jumps and their actuarial and financial applications. Springer-Verlag London, London (2013) Google Scholar |
[10] |
Doléans-Dade, C:Quelques applications de la formule de changement de variables pour les semimartin-gales. Z Wahrscheinlichkeitstheorie verw Gebiete. 16, 181-194 (1970) Google Scholar |
[11] |
Duffie, D, Epstein, LG:Stochastic differential utility. Econometrica. 60(2), 353-394 (1992) Google Scholar |
[12] |
Dupire, B:Functional Itô calculus (2009). Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS. Available at SSRN:https://ssrn.com/abstract=1435551 Google Scholar |
[13] |
El Karoui, N, Peng, S, Quenez, M:Backward stochastic differential equations in finance. Math. Financ. 7, 1-71 (1997) Google Scholar |
[14] |
He, S, Wang, J, Yan, J:Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton (1992) Google Scholar |
[15] |
Jacod, J, Shiryaev, AN:Limit theorems for stochastic processes, 2nd edn. Springer-Verlag Berlin Heidelberg, Berlin (2003) Google Scholar |
[16] |
Kromer, E, Overbeck, L:Representation of BSDE-based dynamic risk measures and dynamic capital allocations. Int. J. Theor. Appl. Financ. 17, 1-16 (2014) Google Scholar |
[17] |
Kromer, E, Overbeck, L:Differentiability of BSVIEs and dynamic capital allocations. Int. J. Theor. Appl. Financ. 20, 1-26 (2017) Google Scholar |
[18] |
Kromer, E, Overbeck, L, Röder, J:Feynman Kac for functional jump diffusions with an application to credit value adjustment. Stat. Probab. Lett. 105, 120-129 (2015) Google Scholar |
[19] |
Kromer, E, Overbeck, L, Röder, J:Path-dependent BSDEs with jumps and their connection to PPIDEs. Stochast. Dyn. 17, 1-37 (2017) Google Scholar |
[20] |
Lazrak, A, Quenez, M:A generalized stochastic differential utility. Math. Oper. Res. 28, 154-180 (2003) Google Scholar |
[21] |
Levental, S, Schroder, M, Sinha, S:A simple proof of functional Itô's lemma for semimartingales with an application. Stat. Probab. Lett. 83, 2019-2026 (2013) Google Scholar |
[22] |
Lin, J:Adapted solution of a backward stochastic nonlinear Volterra integral equation. Stoch. Anal. Appl. 20, 165-183 (2002) Google Scholar |
[23] |
Lu, W:Backward stochastic Volterra integral equations associated with a Lévy process and applications (2016). Available at arXiv:https://arxiv.org/abs/1106.6129 Google Scholar |
[24] |
Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 55-61 (1990) Google Scholar |
[25] |
Pardoux, E, Protter, P:Stochastic Volterra equations with anticipating coefficients. Ann. Probab. 18, 1635-1655 (1990) Google Scholar |
[26] |
Peng, S, Wang, F:BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. Sci. China Math. 59, 1-18 (2016) Google Scholar |
[27] |
Protter, P:Volterra equations driven by semimartingales. Ann. Probab. 13, 519-530 (1985) Google Scholar |
[28] |
Ren, Y:On solutions of backward stochastic volterra integral equations with jumps in hilbert spaces. J. Optim. Theory Appl. 144, 319-333 (2010) Google Scholar |
[29] |
Rong, S:On solutions of backward stochastic differential equations with jumps and applications. Stoch. Process. Appl. 66, 209-236 (1997) Google Scholar |
[30] |
Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control. Optim. 32, 1447-1475 (1994). https://doi.org/101137/S0363012992233858 Google Scholar |
[31] |
Wang, F:BSDEs with jumps and path-dependent parabolic integro-differential equations. Chin. Ann. Math. Ser. B. 36, 625-644 (2015) Google Scholar |
[32] |
Wang, T, Yong, J:Comparison theorems for some backward stochastic Volterra integral equations. Stoch. Process. Appl. 125, 1756-1798 (2015) Google Scholar |
[33] |
Wang, Z, Zhang, X:Non-Lipschitz backward stochastic Volterra type equations with jumps. Stochastics and Dynamics. 7(4), 479-496 (2007) Google Scholar |
[34] |
Yong, J:Backward stochastic Volterra integral equations and some related problems. Stoch. Process. Appl. 116, 779-795 (2006) Google Scholar |
[35] |
Yong, J:Continuous-time dynamic risk measures by backward stochastic Volterra integral equations. Appl Anal. 186, 1429-1442 (2007) Google Scholar |
[36] |
Yong, J:Well-posedness and regularity of backward stochastic volterra integral equations. Probab. Theory Relat. Fields. 142, 21-77 (2008) Google Scholar |
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