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Pathdependent 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, 105127 (2005), 
[2] 
Ankirchner, S, Imkeller, P, Dos Reis, G:Classical and variational differentiability of BSDEs with quadratic growth. Electron. J. Probab. 12, 14181453 (2007), 
[3] 
Barles, G, Buckdahn, R, Pardoux, E:Backward stochastic differential equations and integralpartial differential equations. Stochastics Stochastics Rep. 60, 5783 (1997), 
[4] 
Becherer, D:Bounded solutions to bsdes with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16, 20272054 (2006), 
[5] 
Berger, M, Mizel, V:Volterra equations with itô integrals. J Integr. Equ. 2, 187245 (1980), 
[6] 
Carmona, R:Indifference Pricing:Theory and Applications. Princeton University Press, Princeton (2008), 
[7] 
Cont, R, Fournié, DA:Change of variable formulas for nonanticipative functionals on path space. J. Funct. Anal. 259, 10431072 (2010), 
[8] 
Crépey, S:Financial Modeling, a backward stochastic differential equations perspective. SpringerVerlag Berlin Heidelberg (2013), 
[9] 
Delong, L:Backward stochastic differential equations with jumps and their actuarial and financial applications. SpringerVerlag London, London (2013), 
[10] 
DoléansDade, C:Quelques applications de la formule de changement de variables pour les semimartingales. Z Wahrscheinlichkeitstheorie verw Gebiete. 16, 181194 (1970), 
[11] 
Duffie, D, Epstein, LG:Stochastic differential utility. Econometrica. 60(2), 353394 (1992), 
[12] 
Dupire, B:Functional Itô calculus (2009). Bloomberg Portfolio Research Paper No. 200904FRONTIERS. Available at SSRN:https://ssrn.com/abstract=1435551, 
[13] 
El Karoui, N, Peng, S, Quenez, M:Backward stochastic differential equations in finance. Math. Financ. 7, 171 (1997), 
[14] 
He, S, Wang, J, Yan, J:Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton (1992), 
[15] 
Jacod, J, Shiryaev, AN:Limit theorems for stochastic processes, 2nd edn. SpringerVerlag Berlin Heidelberg, Berlin (2003), 
[16] 
Kromer, E, Overbeck, L:Representation of BSDEbased dynamic risk measures and dynamic capital allocations. Int. J. Theor. Appl. Financ. 17, 116 (2014), 
[17] 
Kromer, E, Overbeck, L:Differentiability of BSVIEs and dynamic capital allocations. Int. J. Theor. Appl. Financ. 20, 126 (2017), 
[18] 
Kromer, E, Overbeck, L, Röder, J:Feynman Kac for functional jump diffusions with an application to credit value adjustment. Stat. Probab. Lett. 105, 120129 (2015), 
[19] 
Kromer, E, Overbeck, L, Röder, J:Pathdependent BSDEs with jumps and their connection to PPIDEs. Stochast. Dyn. 17, 137 (2017), 
[20] 
Lazrak, A, Quenez, M:A generalized stochastic differential utility. Math. Oper. Res. 28, 154180 (2003), 
[21] 
Levental, S, Schroder, M, Sinha, S:A simple proof of functional Itô's lemma for semimartingales with an application. Stat. Probab. Lett. 83, 20192026 (2013), 
[22] 
Lin, J:Adapted solution of a backward stochastic nonlinear Volterra integral equation. Stoch. Anal. Appl. 20, 165183 (2002), 
[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, 
[24] 
Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 5561 (1990), 
[25] 
Pardoux, E, Protter, P:Stochastic Volterra equations with anticipating coefficients. Ann. Probab. 18, 16351655 (1990), 
[26] 
Peng, S, Wang, F:BSDE, pathdependent PDE and nonlinear FeynmanKac formula. Sci. China Math. 59, 118 (2016), 
[27] 
Protter, P:Volterra equations driven by semimartingales. Ann. Probab. 13, 519530 (1985), 
[28] 
Ren, Y:On solutions of backward stochastic volterra integral equations with jumps in hilbert spaces. J. Optim. Theory Appl. 144, 319333 (2010), 
[29] 
Rong, S:On solutions of backward stochastic differential equations with jumps and applications. Stoch. Process. Appl. 66, 209236 (1997), 
[30] 
Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control. Optim. 32, 14471475 (1994). https://doi.org/101137/S0363012992233858, 
[31] 
Wang, F:BSDEs with jumps and pathdependent parabolic integrodifferential equations. Chin. Ann. Math. Ser. B. 36, 625644 (2015), 
[32] 
Wang, T, Yong, J:Comparison theorems for some backward stochastic Volterra integral equations. Stoch. Process. Appl. 125, 17561798 (2015), 
[33] 
Wang, Z, Zhang, X:NonLipschitz backward stochastic Volterra type equations with jumps. Stochastics and Dynamics. 7(4), 479496 (2007), 
[34] 
Yong, J:Backward stochastic Volterra integral equations and some related problems. Stoch. Process. Appl. 116, 779795 (2006), 
[35] 
Yong, J:Continuoustime dynamic risk measures by backward stochastic Volterra integral equations. Appl Anal. 186, 14291442 (2007), 
[36] 
Yong, J:Wellposedness and regularity of backward stochastic volterra integral equations. Probab. Theory Relat. Fields. 142, 2177 (2008), 
show all references
References:
[1] 
Aman, A, N'Zi, M:Backward stochastic nonlinear Volterra integral equations with local Lipschitz drift. Probab. Math. Stat. 25, 105127 (2005), 
[2] 
Ankirchner, S, Imkeller, P, Dos Reis, G:Classical and variational differentiability of BSDEs with quadratic growth. Electron. J. Probab. 12, 14181453 (2007), 
[3] 
Barles, G, Buckdahn, R, Pardoux, E:Backward stochastic differential equations and integralpartial differential equations. Stochastics Stochastics Rep. 60, 5783 (1997), 
[4] 
Becherer, D:Bounded solutions to bsdes with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16, 20272054 (2006), 
[5] 
Berger, M, Mizel, V:Volterra equations with itô integrals. J Integr. Equ. 2, 187245 (1980), 
[6] 
Carmona, R:Indifference Pricing:Theory and Applications. Princeton University Press, Princeton (2008), 
[7] 
Cont, R, Fournié, DA:Change of variable formulas for nonanticipative functionals on path space. J. Funct. Anal. 259, 10431072 (2010), 
[8] 
Crépey, S:Financial Modeling, a backward stochastic differential equations perspective. SpringerVerlag Berlin Heidelberg (2013), 
[9] 
Delong, L:Backward stochastic differential equations with jumps and their actuarial and financial applications. SpringerVerlag London, London (2013), 
[10] 
DoléansDade, C:Quelques applications de la formule de changement de variables pour les semimartingales. Z Wahrscheinlichkeitstheorie verw Gebiete. 16, 181194 (1970), 
[11] 
Duffie, D, Epstein, LG:Stochastic differential utility. Econometrica. 60(2), 353394 (1992), 
[12] 
Dupire, B:Functional Itô calculus (2009). Bloomberg Portfolio Research Paper No. 200904FRONTIERS. Available at SSRN:https://ssrn.com/abstract=1435551, 
[13] 
El Karoui, N, Peng, S, Quenez, M:Backward stochastic differential equations in finance. Math. Financ. 7, 171 (1997), 
[14] 
He, S, Wang, J, Yan, J:Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton (1992), 
[15] 
Jacod, J, Shiryaev, AN:Limit theorems for stochastic processes, 2nd edn. SpringerVerlag Berlin Heidelberg, Berlin (2003), 
[16] 
Kromer, E, Overbeck, L:Representation of BSDEbased dynamic risk measures and dynamic capital allocations. Int. J. Theor. Appl. Financ. 17, 116 (2014), 
[17] 
Kromer, E, Overbeck, L:Differentiability of BSVIEs and dynamic capital allocations. Int. J. Theor. Appl. Financ. 20, 126 (2017), 
[18] 
Kromer, E, Overbeck, L, Röder, J:Feynman Kac for functional jump diffusions with an application to credit value adjustment. Stat. Probab. Lett. 105, 120129 (2015), 
[19] 
Kromer, E, Overbeck, L, Röder, J:Pathdependent BSDEs with jumps and their connection to PPIDEs. Stochast. Dyn. 17, 137 (2017), 
[20] 
Lazrak, A, Quenez, M:A generalized stochastic differential utility. Math. Oper. Res. 28, 154180 (2003), 
[21] 
Levental, S, Schroder, M, Sinha, S:A simple proof of functional Itô's lemma for semimartingales with an application. Stat. Probab. Lett. 83, 20192026 (2013), 
[22] 
Lin, J:Adapted solution of a backward stochastic nonlinear Volterra integral equation. Stoch. Anal. Appl. 20, 165183 (2002), 
[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, 
[24] 
Pardoux, E, Peng, S:Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14, 5561 (1990), 
[25] 
Pardoux, E, Protter, P:Stochastic Volterra equations with anticipating coefficients. Ann. Probab. 18, 16351655 (1990), 
[26] 
Peng, S, Wang, F:BSDE, pathdependent PDE and nonlinear FeynmanKac formula. Sci. China Math. 59, 118 (2016), 
[27] 
Protter, P:Volterra equations driven by semimartingales. Ann. Probab. 13, 519530 (1985), 
[28] 
Ren, Y:On solutions of backward stochastic volterra integral equations with jumps in hilbert spaces. J. Optim. Theory Appl. 144, 319333 (2010), 
[29] 
Rong, S:On solutions of backward stochastic differential equations with jumps and applications. Stoch. Process. Appl. 66, 209236 (1997), 
[30] 
Tang, S, Li, X:Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control. Optim. 32, 14471475 (1994). https://doi.org/101137/S0363012992233858, 
[31] 
Wang, F:BSDEs with jumps and pathdependent parabolic integrodifferential equations. Chin. Ann. Math. Ser. B. 36, 625644 (2015), 
[32] 
Wang, T, Yong, J:Comparison theorems for some backward stochastic Volterra integral equations. Stoch. Process. Appl. 125, 17561798 (2015), 
[33] 
Wang, Z, Zhang, X:NonLipschitz backward stochastic Volterra type equations with jumps. Stochastics and Dynamics. 7(4), 479496 (2007), 
[34] 
Yong, J:Backward stochastic Volterra integral equations and some related problems. Stoch. Process. Appl. 116, 779795 (2006), 
[35] 
Yong, J:Continuoustime dynamic risk measures by backward stochastic Volterra integral equations. Appl Anal. 186, 14291442 (2007), 
[36] 
Yong, J:Wellposedness and regularity of backward stochastic volterra integral equations. Probab. Theory Relat. Fields. 142, 2177 (2008), 
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