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

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

Ludger Overbeck and  Jasmin A. L. Röder ,

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.
Keywords: Path-dependent backward stochastic Volterra integral equation, Jump diffusion, Path-differentiability, Duality principle, Comparison theorem, Functional Itô formula, Dynamic coherent risk measure.
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),

[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),

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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