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 Published  June 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) 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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