Existence, uniqueness and strict comparison theorems for BSDEs driven by RCLL martingales

Tianyang Nie; Marek Rutkowski

doi:10.3934/puqr.2021016

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December 2021, 6(4): 319-342. doi: 10.3934/puqr.2021016

Existence, uniqueness and strict comparison theorems for BSDEs driven by RCLL martingales

Tianyang Nie ^1,, and Marek Rutkowski ^2,3,

1.	School of Mathematics, Shandong University, Jinan 250100, Shandong, China
2.	School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
3.	Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-661 Warszawa, Poland

E-mail: nietianyang@sdu.edu.cn (Tianyang NIE)

Received March 25, 2021 Accepted November 30, 2021 Published December 2021

Fund Project: The research of M. Rutkowski was supported by the Australian Research Council Discovery Project (Grant No. DP200101550). The work of T. Nie was supported by the National Natural Science Foundation of China (Grant Nos. 12022108, 11971267, 11831010, 61961160732) and Natural Science Foundation of Shandong Province (Grant Nos. ZR2019ZD42, ZR2020ZD24)

The existence, uniqueness, and strict comparison for solutions to a BSDE driven by a multi-dimensional RCLL martingale are developed. The goal is to develop a general multi-asset framework encompassing a wide spectrum of non-linear financial models with jumps, including as particular cases, the setups studied by Peng and Xu [27, 28] and Dumitrescu et al. [7] who dealt with BSDEs driven by a one-dimensional Brownian motion and a purely discontinuous martingale with a single jump.

Keywords: Backward stochastic differential equation, RCLL martingale, Comparison theorem.

Mathematics Subject Classification: 60H10, 60H30, 91G30, 91G40.

Citation: Tianyang Nie, Marek Rutkowski. Existence, uniqueness and strict comparison theorems for BSDEs driven by RCLL martingales. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 319-342. doi: 10.3934/puqr.2021016

References:

[1]	Barles, G., Buckdahn, R. and Pardoux, E., Backward stochastic differential equations and integral-partial differential equations, Stochastics and Stochastic Reports, 1997, 60(1–2): 57−83.
[2]	Bielecki, T. R., Cialenco, I. and Rutkowski, M., Arbitrage-free pricing of derivatives in nonlinear market models,Probability, Uncertainty and Quantitative Risk, 2018, 3: 2, doi: 10.1186/s41546-018-0027-x.
[3]	Bielecki, T. R., Jeanblanc, M. and Rutkowski, M., Credit Risk Modeling, Osaka University Press, Osaka, 2009.
[4]	Carbone, R., Ferrario, B. and Santacroce, M., Backward stochastic differential equations driven by càdlàg martingales, Theory of Probability & Its Applications, 2008, 52(2): 304−314.
[5]	Cohen, S. N. and Elliott, R. J., Stochastic Calculus and Applications, Springer, New York, 2015.
[6]	Dumitrescu, R., Grigorova, M., Quenez, M. C. and Sulem, A., BSDEs with Default Jump, In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K. and Munthe-Kaas, H. (eds.), Computation and Combinatorics in Dynamics, Stochastics and Control, Abel Symposia, Springer, Cham, 2018.
[7]	Dumitrescu, R., Quenez, M. C. and Sulem, A., American options in an imperfect complete market with default, ESAIM: Proceedings and Surveys, 2018, 64: 93−110. doi: 10.1051/proc/201864093.
[8]	El Karoui, N. and Huang, S., A general result of existence and uniqueness of backward stochastic differential equations. In: El Karoui, N. and Mazliak, L. (eds.), Backward Stochastic Differential Equations, Pitman Research Notes in Mathematics Series, Addison Wesley Longman Ltd., Harlow, Essex, 1997, 364: 27–36.
[9]	El Karoui, N., Matoussi, A. and Ngoupeyou, A., Quadratic exponential semimartingales and application to BSDEs with jumps, arXiv: 1603.06191, 2016.
[10]	El Karoui, N., Peng, S. and Quenez, M. C., Backward stochastic differential equations in finance, Mathematical Finance, 1997, 7(1): 1−71. doi: 10.1111/1467-9965.00022.
[11]	He, S., Wang, J. and Yan, J., Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992.
[12]	Jacod, J. and Shiryaev, A. N., Limit Theorems for Stochastic Processes, 2nd ed., Springer, Berlin, 2003.
[13]	Jeanblanc, M. and Le Cam, Y., Immersion property and credit risk modelling, In: Delbaen, F., Rasonyi, M. and Stricker, C.(eds.), Optimality and Risk-Modern Trends in Mathematical Finance: The Kabanov Festschrift, Springer, Berlin, 2009.
[14]	Jeanblanc, M., Matoussi, A. and Ngoupeyou, A., Robust utility maximization problem in a discontinuous filtration, arXiv: 1201.2690v3, 2013.
[15]	Kim, E., Nie, T. and Rutkowski, M., American options in nonlinear markets, Electronic Journal of Probability, 2021, 26: 1−41.
[16]	Kim, E., Nie, T. and Rutkowski, M., Arbitrage-free pricing of game options in nonlinear markets, arXiv: 1807.05448v1, 2018.
[17]	Kusuoka, S., A remark on default risk models, Advances in Mathematical Economics, 1999, 1: 69−82.
[18]	Li, J., Fully coupled forward-backward stochastic differential equations with general martingale, Acta Mathematica Scientia, 2006, 26(3): 443−450. doi: 10.1016/S0252-9602(06)60068-4.
[19]	Morlais, M., Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem, Finance and Stochastics, 2009, 13(1): 121−150. doi: 10.1007/s00780-008-0079-3.
[20]	Nie, T. and Rutkowski, M., BSDEs driven by multidimensional martingales and their applications to markets with funding costs, Theory of Probability & Its Applications, 2016, 60(4): 604−630.
[21]	Nie, T. and Rutkowski, M., Fair bilateral pricing under funding costs and exogenous collateralization, Mathematical Finance, 2018, 28(2): 621−655. doi: 10.1111/mafi.12145.
[22]	Nie, T. and Rutkowski, M., Reflected BSDEs and doubly reflected BSDEs driven by RCLL martingales, Stochastics and Dynamics, 2021, https://doi.org/10.1142/S0219493722500125.
[23]	Papapantoleon, T., Possamaï, D. and Saplaouras, A., Existence and uniqueness results for BSDEs with jumps: the whole nine yards, Electronic Journal of Probability, 2018, 23: 1−68.
[24]	Peng, S. and Xu, X., BSDEs with random default time and their applications to default risk, arXiv: 0910.2091, 2009.
[25]	Peng, S. and Xu, X., BSDEs with random default time and related zero-sum stochastic differential games, Comptes Rendus Mathematique, 2010, 348(3–4): 193−198.
[26]	Protter, P. E., Stochastic Integration and Differential Equations, 2nd ed., Springer, Berlin, 2004.
[27]	Quenez, M. C. and Sulem, A., BSDEs with jumps, optimization and applications to dynamic risk measures, Stochastic Processes and their Applications, 2013, 123(8): 3328−3357. doi: 10.1016/j.spa.2013.02.016.
[28]	Royer, M., Backward stochastic differential equations with jumps and related non-linear expectations, Stochastic Processes and their Applications, 2006, 116(10): 1358−1376. doi: 10.1016/j.spa.2006.02.009.
[29]	Tang, S. and Li, X., Necessary conditions for optimal control of stochastic systems with random jumps, SIAM Journal on Control and Optimization, 1994, 32(5): 1447−1475. doi: 10.1137/S0363012992233858.

show all references

References:

[1]	Barles, G., Buckdahn, R. and Pardoux, E., Backward stochastic differential equations and integral-partial differential equations, Stochastics and Stochastic Reports, 1997, 60(1–2): 57−83.
[2]	Bielecki, T. R., Cialenco, I. and Rutkowski, M., Arbitrage-free pricing of derivatives in nonlinear market models,Probability, Uncertainty and Quantitative Risk, 2018, 3: 2, doi: 10.1186/s41546-018-0027-x.
[3]	Bielecki, T. R., Jeanblanc, M. and Rutkowski, M., Credit Risk Modeling, Osaka University Press, Osaka, 2009.
[4]	Carbone, R., Ferrario, B. and Santacroce, M., Backward stochastic differential equations driven by càdlàg martingales, Theory of Probability & Its Applications, 2008, 52(2): 304−314.
[5]	Cohen, S. N. and Elliott, R. J., Stochastic Calculus and Applications, Springer, New York, 2015.
[6]	Dumitrescu, R., Grigorova, M., Quenez, M. C. and Sulem, A., BSDEs with Default Jump, In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K. and Munthe-Kaas, H. (eds.), Computation and Combinatorics in Dynamics, Stochastics and Control, Abel Symposia, Springer, Cham, 2018.
[7]	Dumitrescu, R., Quenez, M. C. and Sulem, A., American options in an imperfect complete market with default, ESAIM: Proceedings and Surveys, 2018, 64: 93−110. doi: 10.1051/proc/201864093.
[8]	El Karoui, N. and Huang, S., A general result of existence and uniqueness of backward stochastic differential equations. In: El Karoui, N. and Mazliak, L. (eds.), Backward Stochastic Differential Equations, Pitman Research Notes in Mathematics Series, Addison Wesley Longman Ltd., Harlow, Essex, 1997, 364: 27–36.
[9]	El Karoui, N., Matoussi, A. and Ngoupeyou, A., Quadratic exponential semimartingales and application to BSDEs with jumps, arXiv: 1603.06191, 2016.
[10]	El Karoui, N., Peng, S. and Quenez, M. C., Backward stochastic differential equations in finance, Mathematical Finance, 1997, 7(1): 1−71. doi: 10.1111/1467-9965.00022.
[11]	He, S., Wang, J. and Yan, J., Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992.
[12]	Jacod, J. and Shiryaev, A. N., Limit Theorems for Stochastic Processes, 2nd ed., Springer, Berlin, 2003.
[13]	Jeanblanc, M. and Le Cam, Y., Immersion property and credit risk modelling, In: Delbaen, F., Rasonyi, M. and Stricker, C.(eds.), Optimality and Risk-Modern Trends in Mathematical Finance: The Kabanov Festschrift, Springer, Berlin, 2009.
[14]	Jeanblanc, M., Matoussi, A. and Ngoupeyou, A., Robust utility maximization problem in a discontinuous filtration, arXiv: 1201.2690v3, 2013.
[15]	Kim, E., Nie, T. and Rutkowski, M., American options in nonlinear markets, Electronic Journal of Probability, 2021, 26: 1−41.
[16]	Kim, E., Nie, T. and Rutkowski, M., Arbitrage-free pricing of game options in nonlinear markets, arXiv: 1807.05448v1, 2018.
[17]	Kusuoka, S., A remark on default risk models, Advances in Mathematical Economics, 1999, 1: 69−82.
[18]	Li, J., Fully coupled forward-backward stochastic differential equations with general martingale, Acta Mathematica Scientia, 2006, 26(3): 443−450. doi: 10.1016/S0252-9602(06)60068-4.
[19]	Morlais, M., Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem, Finance and Stochastics, 2009, 13(1): 121−150. doi: 10.1007/s00780-008-0079-3.
[20]	Nie, T. and Rutkowski, M., BSDEs driven by multidimensional martingales and their applications to markets with funding costs, Theory of Probability & Its Applications, 2016, 60(4): 604−630.
[21]	Nie, T. and Rutkowski, M., Fair bilateral pricing under funding costs and exogenous collateralization, Mathematical Finance, 2018, 28(2): 621−655. doi: 10.1111/mafi.12145.
[22]	Nie, T. and Rutkowski, M., Reflected BSDEs and doubly reflected BSDEs driven by RCLL martingales, Stochastics and Dynamics, 2021, https://doi.org/10.1142/S0219493722500125.
[23]	Papapantoleon, T., Possamaï, D. and Saplaouras, A., Existence and uniqueness results for BSDEs with jumps: the whole nine yards, Electronic Journal of Probability, 2018, 23: 1−68.
[24]	Peng, S. and Xu, X., BSDEs with random default time and their applications to default risk, arXiv: 0910.2091, 2009.
[25]	Peng, S. and Xu, X., BSDEs with random default time and related zero-sum stochastic differential games, Comptes Rendus Mathematique, 2010, 348(3–4): 193−198.
[26]	Protter, P. E., Stochastic Integration and Differential Equations, 2nd ed., Springer, Berlin, 2004.
[27]	Quenez, M. C. and Sulem, A., BSDEs with jumps, optimization and applications to dynamic risk measures, Stochastic Processes and their Applications, 2013, 123(8): 3328−3357. doi: 10.1016/j.spa.2013.02.016.
[28]	Royer, M., Backward stochastic differential equations with jumps and related non-linear expectations, Stochastic Processes and their Applications, 2006, 116(10): 1358−1376. doi: 10.1016/j.spa.2006.02.009.
[29]	Tang, S. and Li, X., Necessary conditions for optimal control of stochastic systems with random jumps, SIAM Journal on Control and Optimization, 1994, 32(5): 1447−1475. doi: 10.1137/S0363012992233858.

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American Institute of Mathematical Sciences

Existence, uniqueness and strict comparison theorems for BSDEs driven by RCLL martingales

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