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January  2018, 3: 9 doi: 10.1186/s41546-018-0034-y

Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting

Christel Geiss 1, and  Alexander Steinicke 2,,

1. University of Jyvaskyla, Department of Mathematics and Statistics, P. O. Box 35, 40014 Jyvaskyla, Finland;

2. Department of Mathematics and Information Technology, Montanuniversitaet Leoben, Leoben, Austria

Received  January 04, 2018 Revised  November 29, 2018 Published  December 2018

We show that the comparison results for a backward SDE with jumps established in Royer (Stoch. Process. Appl 116:1358-1376, 2006) and Yin and Mao (J. Math. Anal. Appl 346:345-358, 2008) hold under more simplified conditions. Moreover, we prove existence and uniqueness allowing the coefficients in the linear growth- and monotonicity-condition for the generator to be random and time-dependent. In the L2-case with linear growth, this also generalizes the results of Kruse and Popier (Stochastics 88:491-539, 2016). For the proof of the comparison result, we introduce an approximation technique:Given a BSDE driven by Brownian motion and Poisson random measure, we approximate it by BSDEs where the Poisson random measure admits only jumps of size larger than 1/n.
Keywords: Backward stochastic differential equation, Lévy process, comparison theorem, existence and uniqueness.
Mathematics Subject Classification: 60H10.
Citation: Christel Geiss, Alexander Steinicke. Existence, uniqueness and comparison results for BSDEs with Lévy jumps in an extended monotonic generator setting. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 9-. doi: 10.1186/s41546-018-0034-y
References:
[1]

Applebaum, D:Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)

[2]

Barles, G, Buckdahn, R, Pardoux, É:Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60(1-2), 57-83 (1997)

[3]

Becherer, D, Büttner, M, Kentia, K:On the monotone stability approach to BSDEs with jumps:Extensions, concrete criteria and examples (2018). https://arxiv.org/abs/1607.06644

[4]

Billingsley, P:Convergence of probability measures. Wiley, New York (1968)

[5]

Briand, P, Delyon, B, Hu, Y, Pardoux, E, Stoica, L:Lp solutions of backward stochastic differential equations. Stoch. Proc. Appl. 108, 109-129 (2003)

[6]

Cao, Z, Yan, J:A comparison theorem for solutions of backward stochastic differential equations. Adv. Math. 28(4), 304-308 (1999)

[7]

Cohen, S, Elliott, R, Pearce, C:A General Comparison Theorem for Backward Stochastic Differential Equations. Adv. Appl. Probab. 42(3), 878-898 (2010)

[8]

Darling, R, Pardoux, É:Backwards sde with random terminal time and applications to semilinear elliptic pde. Ann. Probab. 25(3), 1135-1159 (1997)

[9]

Delzeith, O:On Skorohod spaces as universal sample path spaces (2004). https://arxiv.org/abs/math/0412092v1

[10]

El Karoui, N, Hamadène, S, Matoussi, A:Backward stochastic differential equations. In:Carmona, R (ed.)

[11]

Indifference Hedging:Theory and Applications, pp. 267-320. Princeton University Press (2009)

[12]

El Karoui, N, Tan, X:Capacities, measurable selection and dynamic programming part I:abstract framework (2013). https://arxiv.org/abs/1310.3363

[13]

El Karoui, N, Peng, S, Quenez, M:Backward Stochastic Differential Equations in Finance. Math. Financ. 7(1), 1-71 (1997)

[14]

Fan, S, Jiang, L:A Generalized Comparison Theorem for BSDEs and Its Applications. J. Theor. Probab. 25, 50-61 (2012)

[15]

Geiss, C, Steinicke, A:Existence, Uniqueness and Malliavin Differentiability of Lévy-driven BSDEs with locally Lipschitz Driver (2018). https://arxiv.org/abs/1805.05851

[16]

Geiss, S, Ylinen, J:Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs(2018). is to appear in:Memoirs AMS, 2018

[17]

Gobet, E, Turkedjiev, P:Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions. Math. Comp. 85, 1359-1391 (2016)

[18]

He, SW, Wang, JG, Yan, JA:Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton(1992)

[19]

Kechris, A:Classical Descriptive Set Theory. Springer, New York (1994)

[20]

Kruse, T, Popier, A:BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration. Stochastics. 88(4), 491-539 (2016)

[21]

Mao, X:Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients. Stoch. Process. Appl. 58, 281-292 (1995)

[22]

Mao, X:Stochastic Differential Equations and Applications. Woodhead Publishing Limited, Cambridge(1997)

[23]

Meyer, PA:Une remarque sur le calcul stochastique dépendant d'un paramètre. Séminaire probabilités(Strasbourg), tome. 13, 199-203 (1979)

[24]

Pardoux, É:Generalized discontinuous backward stochastic differential equations. In:El Karoui, N, Mazliak, L (eds.) Backward Stochastic Differential Equations, Pitman Res. Notes Math., vol. 364, pp. 207-219. Longman, Harlow (1997)

[25]

Pardoux, É, Zhang, S:Generalized BSDEs and nonlinear Neumann boundary value problems. Probab. Theory Relat. Fields. 110, 535-558 (1996)

[26]

Peng, S:A generalized dynamic programming principle and hamilton-jacobi-bellman equation. Stoch. Stoch. Rep. 38, 119-134 (1992)

[27]

Peng, S:Backward stochastic differential equation, nonlinear expectation and their applications. Proceedings of the International Congress of Mathematicians, Volume I, pp. 393-432. Hindustan Book Agency, New Delhi (2010)

[28]

Protter, P:Stochastic Integration and Differential Equations. Springer, Berlin (2004)

[29]

Royer, M:Backward stochastic differential equations with jumps and related non-linear expectations. Stoch. Process. Appl. 116(10), 1358-1376 (2006)

[30]

Sato, K:Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge(1999)

[31]

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

[32]

Steinicke, A:Functionals of a Lévy Process on Canonical and Generic Probability Spaces. J. Theoret. Probab. 29, 443-458 (2016)

[33]

Sow, AB:BSDE with jumps and non-Lipschitz coefficients:Application to large deviations. Braz. J. Probab. Stat. 28(1), 96-108 (2014)

[34]

Yao, S:Lp-solutions of Backward Stochastic Differential Equations with Jumps. Stoch. Proc. Appl. 127(11), 3465-3511 (2017)

[35]

Yin, J, Mao, X:The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications. J. Math. Anal. Appl. 346, 345-358 (2008)

[36]

Ylinen, J:Weighted Bounded Mean Oscillation applied to Backward Stochastic Differential Equations(2017). https://arxiv.org/abs/1501.01183

show all references

References:
[1]

Applebaum, D:Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)

[2]

Barles, G, Buckdahn, R, Pardoux, É:Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60(1-2), 57-83 (1997)

[3]

Becherer, D, Büttner, M, Kentia, K:On the monotone stability approach to BSDEs with jumps:Extensions, concrete criteria and examples (2018). https://arxiv.org/abs/1607.06644

[4]

Billingsley, P:Convergence of probability measures. Wiley, New York (1968)

[5]

Briand, P, Delyon, B, Hu, Y, Pardoux, E, Stoica, L:Lp solutions of backward stochastic differential equations. Stoch. Proc. Appl. 108, 109-129 (2003)

[6]

Cao, Z, Yan, J:A comparison theorem for solutions of backward stochastic differential equations. Adv. Math. 28(4), 304-308 (1999)

[7]

Cohen, S, Elliott, R, Pearce, C:A General Comparison Theorem for Backward Stochastic Differential Equations. Adv. Appl. Probab. 42(3), 878-898 (2010)

[8]

Darling, R, Pardoux, É:Backwards sde with random terminal time and applications to semilinear elliptic pde. Ann. Probab. 25(3), 1135-1159 (1997)

[9]

Delzeith, O:On Skorohod spaces as universal sample path spaces (2004). https://arxiv.org/abs/math/0412092v1

[10]

El Karoui, N, Hamadène, S, Matoussi, A:Backward stochastic differential equations. In:Carmona, R (ed.)

[11]

Indifference Hedging:Theory and Applications, pp. 267-320. Princeton University Press (2009)

[12]

El Karoui, N, Tan, X:Capacities, measurable selection and dynamic programming part I:abstract framework (2013). https://arxiv.org/abs/1310.3363

[13]

El Karoui, N, Peng, S, Quenez, M:Backward Stochastic Differential Equations in Finance. Math. Financ. 7(1), 1-71 (1997)

[14]

Fan, S, Jiang, L:A Generalized Comparison Theorem for BSDEs and Its Applications. J. Theor. Probab. 25, 50-61 (2012)

[15]

Geiss, C, Steinicke, A:Existence, Uniqueness and Malliavin Differentiability of Lévy-driven BSDEs with locally Lipschitz Driver (2018). https://arxiv.org/abs/1805.05851

[16]

Geiss, S, Ylinen, J:Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs(2018). is to appear in:Memoirs AMS, 2018

[17]

Gobet, E, Turkedjiev, P:Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions. Math. Comp. 85, 1359-1391 (2016)

[18]

He, SW, Wang, JG, Yan, JA:Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton(1992)

[19]

Kechris, A:Classical Descriptive Set Theory. Springer, New York (1994)

[20]

Kruse, T, Popier, A:BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration. Stochastics. 88(4), 491-539 (2016)

[21]

Mao, X:Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients. Stoch. Process. Appl. 58, 281-292 (1995)

[22]

Mao, X:Stochastic Differential Equations and Applications. Woodhead Publishing Limited, Cambridge(1997)

[23]

Meyer, PA:Une remarque sur le calcul stochastique dépendant d'un paramètre. Séminaire probabilités(Strasbourg), tome. 13, 199-203 (1979)

[24]

Pardoux, É:Generalized discontinuous backward stochastic differential equations. In:El Karoui, N, Mazliak, L (eds.) Backward Stochastic Differential Equations, Pitman Res. Notes Math., vol. 364, pp. 207-219. Longman, Harlow (1997)

[25]

Pardoux, É, Zhang, S:Generalized BSDEs and nonlinear Neumann boundary value problems. Probab. Theory Relat. Fields. 110, 535-558 (1996)

[26]

Peng, S:A generalized dynamic programming principle and hamilton-jacobi-bellman equation. Stoch. Stoch. Rep. 38, 119-134 (1992)

[27]

Peng, S:Backward stochastic differential equation, nonlinear expectation and their applications. Proceedings of the International Congress of Mathematicians, Volume I, pp. 393-432. Hindustan Book Agency, New Delhi (2010)

[28]

Protter, P:Stochastic Integration and Differential Equations. Springer, Berlin (2004)

[29]

Royer, M:Backward stochastic differential equations with jumps and related non-linear expectations. Stoch. Process. Appl. 116(10), 1358-1376 (2006)

[30]

Sato, K:Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge(1999)

[31]

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

[32]

Steinicke, A:Functionals of a Lévy Process on Canonical and Generic Probability Spaces. J. Theoret. Probab. 29, 443-458 (2016)

[33]

Sow, AB:BSDE with jumps and non-Lipschitz coefficients:Application to large deviations. Braz. J. Probab. Stat. 28(1), 96-108 (2014)

[34]

Yao, S:Lp-solutions of Backward Stochastic Differential Equations with Jumps. Stoch. Proc. Appl. 127(11), 3465-3511 (2017)

[35]

Yin, J, Mao, X:The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications. J. Math. Anal. Appl. 346, 345-358 (2008)

[36]

Ylinen, J:Weighted Bounded Mean Oscillation applied to Backward Stochastic Differential Equations(2017). https://arxiv.org/abs/1501.01183

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