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

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.
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) Google Scholar

[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) Google Scholar

[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 Google Scholar

[4]

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

[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) Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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 Google Scholar

[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 Google Scholar

[17]

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

[18]

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

[19]

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

[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) Google Scholar

[21]

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

[22]

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

[23]

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

[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) Google Scholar

[25]

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

[26]

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

[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) Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[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) Google Scholar

[36]

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

show all references

References:
[1]

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

[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) Google Scholar

[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 Google Scholar

[4]

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

[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) Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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 Google Scholar

[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 Google Scholar

[17]

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

[18]

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

[19]

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

[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) Google Scholar

[21]

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

[22]

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

[23]

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

[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) Google Scholar

[25]

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

[26]

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

[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) Google Scholar

[28]

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

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[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) Google Scholar

[36]

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

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