January  2018, 3: 4 doi: 10.1186/s41546-018-0030-2

Path-dependent backward stochastic Volterra integral equations with jumps, differentiability and duality principle

Institute of Mathematics, University of Gießen, Arndtsraße 2, 35392 Gießen, Germany

Received  May 30, 2017 Revised  April 23, 2018

Fund Project: The authors thank the editor and two anonymous referees for their helpful suggestions.

We study the existence and uniqueness of a solution to path-dependent backward stochastic Volterra integral equations (BSVIEs) with jumps, where pathdependence means the dependence of the free term and generator of a path of a càdlàg process. Furthermore, we prove path-differentiability of such a solution and establish the duality principle between a linear path-dependent forward stochastic Volterra integral equation (FSVIE) with jumps and a linear path-dependent BSVIE with jumps. As a result of the duality principle we get a comparison theorem and derive a class of dynamic coherent risk measures based on path-dependent BSVIEs with jumps.
Citation: Ludger Overbeck, Jasmin A. L. Röder. Path-dependent backward stochastic Volterra integral equations with jumps, differentiability and duality principle. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 4-. doi: 10.1186/s41546-018-0030-2
References:
[1]

Aman, A, N'Zi, M:Backward stochastic nonlinear Volterra integral equations with local Lipschitz drift. Probab. Math. Stat. 25, 105-127 (2005),

[2]

Ankirchner, S, Imkeller, P, Dos Reis, G:Classical and variational differentiability of BSDEs with quadratic growth. Electron. J. Probab. 12, 1418-1453 (2007),

[3]

Barles, G, Buckdahn, R, Pardoux, E:Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60, 57-83 (1997),

[4]

Becherer, D:Bounded solutions to bsdes with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16, 2027-2054 (2006),

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Berger, M, Mizel, V:Volterra equations with itô integrals. J Integr. Equ. 2, 187-245 (1980),

[6]

Carmona, R:Indifference Pricing:Theory and Applications. Princeton University Press, Princeton (2008),

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Cont, R, Fournié, DA:Change of variable formulas for non-anticipative functionals on path space. J. Funct. Anal. 259, 1043-1072 (2010),

[8]

Crépey, S:Financial Modeling, a backward stochastic differential equations perspective. Springer-Verlag Berlin Heidelberg (2013),

[9]

Delong, L:Backward stochastic differential equations with jumps and their actuarial and financial applications. Springer-Verlag London, London (2013),

[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),

[11]

Duffie, D, Epstein, LG:Stochastic differential utility. Econometrica. 60(2), 353-394 (1992),

[12]

Dupire, B:Functional Itô calculus (2009). Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS. 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, 1-71 (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. Springer-Verlag Berlin Heidelberg, Berlin (2003),

[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),

[17]

Kromer, E, Overbeck, L:Differentiability of BSVIEs and dynamic capital allocations. Int. J. Theor. Appl. Financ. 20, 1-26 (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, 120-129 (2015),

[19]

Kromer, E, Overbeck, L, Röder, J:Path-dependent BSDEs with jumps and their connection to PPIDEs. Stochast. Dyn. 17, 1-37 (2017),

[20]

Lazrak, A, Quenez, M:A generalized stochastic differential utility. Math. Oper. Res. 28, 154-180 (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, 2019-2026 (2013),

[22]

Lin, J:Adapted solution of a backward stochastic nonlinear Volterra integral equation. Stoch. Anal. Appl. 20, 165-183 (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, 55-61 (1990),

[25]

Pardoux, E, Protter, P:Stochastic Volterra equations with anticipating coefficients. Ann. Probab. 18, 1635-1655 (1990),

[26]

Peng, S, Wang, F:BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. Sci. China Math. 59, 1-18 (2016),

[27]

Protter, P:Volterra equations driven by semimartingales. Ann. Probab. 13, 519-530 (1985),

[28]

Ren, Y:On solutions of backward stochastic volterra integral equations with jumps in hilbert spaces. J. Optim. Theory Appl. 144, 319-333 (2010),

[29]

Rong, S:On solutions of backward stochastic differential equations with jumps and applications. Stoch. Process. Appl. 66, 209-236 (1997),

[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,

[31]

Wang, F:BSDEs with jumps and path-dependent parabolic integro-differential equations. Chin. Ann. Math. Ser. B. 36, 625-644 (2015),

[32]

Wang, T, Yong, J:Comparison theorems for some backward stochastic Volterra integral equations. Stoch. Process. Appl. 125, 1756-1798 (2015),

[33]

Wang, Z, Zhang, X:Non-Lipschitz backward stochastic Volterra type equations with jumps. Stochastics and Dynamics. 7(4), 479-496 (2007),

[34]

Yong, J:Backward stochastic Volterra integral equations and some related problems. Stoch. Process. Appl. 116, 779-795 (2006),

[35]

Yong, J:Continuous-time dynamic risk measures by backward stochastic Volterra integral equations. Appl Anal. 186, 1429-1442 (2007),

[36]

Yong, J:Well-posedness and regularity of backward stochastic volterra integral equations. Probab. Theory Relat. Fields. 142, 21-77 (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, 105-127 (2005),

[2]

Ankirchner, S, Imkeller, P, Dos Reis, G:Classical and variational differentiability of BSDEs with quadratic growth. Electron. J. Probab. 12, 1418-1453 (2007),

[3]

Barles, G, Buckdahn, R, Pardoux, E:Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60, 57-83 (1997),

[4]

Becherer, D:Bounded solutions to bsdes with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16, 2027-2054 (2006),

[5]

Berger, M, Mizel, V:Volterra equations with itô integrals. J Integr. Equ. 2, 187-245 (1980),

[6]

Carmona, R:Indifference Pricing:Theory and Applications. Princeton University Press, Princeton (2008),

[7]

Cont, R, Fournié, DA:Change of variable formulas for non-anticipative functionals on path space. J. Funct. Anal. 259, 1043-1072 (2010),

[8]

Crépey, S:Financial Modeling, a backward stochastic differential equations perspective. Springer-Verlag Berlin Heidelberg (2013),

[9]

Delong, L:Backward stochastic differential equations with jumps and their actuarial and financial applications. Springer-Verlag London, London (2013),

[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),

[11]

Duffie, D, Epstein, LG:Stochastic differential utility. Econometrica. 60(2), 353-394 (1992),

[12]

Dupire, B:Functional Itô calculus (2009). Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS. 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, 1-71 (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. Springer-Verlag Berlin Heidelberg, Berlin (2003),

[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),

[17]

Kromer, E, Overbeck, L:Differentiability of BSVIEs and dynamic capital allocations. Int. J. Theor. Appl. Financ. 20, 1-26 (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, 120-129 (2015),

[19]

Kromer, E, Overbeck, L, Röder, J:Path-dependent BSDEs with jumps and their connection to PPIDEs. Stochast. Dyn. 17, 1-37 (2017),

[20]

Lazrak, A, Quenez, M:A generalized stochastic differential utility. Math. Oper. Res. 28, 154-180 (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, 2019-2026 (2013),

[22]

Lin, J:Adapted solution of a backward stochastic nonlinear Volterra integral equation. Stoch. Anal. Appl. 20, 165-183 (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, 55-61 (1990),

[25]

Pardoux, E, Protter, P:Stochastic Volterra equations with anticipating coefficients. Ann. Probab. 18, 1635-1655 (1990),

[26]

Peng, S, Wang, F:BSDE, path-dependent PDE and nonlinear Feynman-Kac formula. Sci. China Math. 59, 1-18 (2016),

[27]

Protter, P:Volterra equations driven by semimartingales. Ann. Probab. 13, 519-530 (1985),

[28]

Ren, Y:On solutions of backward stochastic volterra integral equations with jumps in hilbert spaces. J. Optim. Theory Appl. 144, 319-333 (2010),

[29]

Rong, S:On solutions of backward stochastic differential equations with jumps and applications. Stoch. Process. Appl. 66, 209-236 (1997),

[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,

[31]

Wang, F:BSDEs with jumps and path-dependent parabolic integro-differential equations. Chin. Ann. Math. Ser. B. 36, 625-644 (2015),

[32]

Wang, T, Yong, J:Comparison theorems for some backward stochastic Volterra integral equations. Stoch. Process. Appl. 125, 1756-1798 (2015),

[33]

Wang, Z, Zhang, X:Non-Lipschitz backward stochastic Volterra type equations with jumps. Stochastics and Dynamics. 7(4), 479-496 (2007),

[34]

Yong, J:Backward stochastic Volterra integral equations and some related problems. Stoch. Process. Appl. 116, 779-795 (2006),

[35]

Yong, J:Continuous-time dynamic risk measures by backward stochastic Volterra integral equations. Appl Anal. 186, 1429-1442 (2007),

[36]

Yong, J:Well-posedness and regularity of backward stochastic volterra integral equations. Probab. Theory Relat. Fields. 142, 21-77 (2008),

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