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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

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.

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. Google Scholar

[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. Google Scholar

[3]

Bielecki, T. R., Jeanblanc, M. and Rutkowski, M., Credit Risk Modeling, Osaka University Press, Osaka, 2009. Google Scholar

[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. Google Scholar

[5]

Cohen, S. N. and Elliott, R. J., Stochastic Calculus and Applications, Springer, New York, 2015. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[9]

El Karoui, N., Matoussi, A. and Ngoupeyou, A., Quadratic exponential semimartingales and application to BSDEs with jumps, arXiv: 1603.06191, 2016. Google Scholar

[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.  Google Scholar

[11]

He, S., Wang, J. and Yan, J., Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992. Google Scholar

[12]

Jacod, J. and Shiryaev, A. N., Limit Theorems for Stochastic Processes, 2nd ed., Springer, Berlin, 2003. Google Scholar

[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. Google Scholar

[14]

Jeanblanc, M., Matoussi, A. and Ngoupeyou, A., Robust utility maximization problem in a discontinuous filtration, arXiv: 1201.2690v3, 2013. Google Scholar

[15]

Kim, E., Nie, T. and Rutkowski, M., American options in nonlinear markets, Electronic Journal of Probability, 2021, 26: 1−41. Google Scholar

[16]

Kim, E., Nie, T. and Rutkowski, M., Arbitrage-free pricing of game options in nonlinear markets, arXiv: 1807.05448v1, 2018. Google Scholar

[17]

Kusuoka, S., A remark on default risk models, Advances in Mathematical Economics, 1999, 1: 69−82. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

Peng, S. and Xu, X., BSDEs with random default time and their applications to default risk, arXiv: 0910.2091, 2009. Google Scholar

[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. Google Scholar

[26]

Protter, P. E., Stochastic Integration and Differential Equations, 2nd ed., Springer, Berlin, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

Bielecki, T. R., Jeanblanc, M. and Rutkowski, M., Credit Risk Modeling, Osaka University Press, Osaka, 2009. Google Scholar

[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. Google Scholar

[5]

Cohen, S. N. and Elliott, R. J., Stochastic Calculus and Applications, Springer, New York, 2015. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[9]

El Karoui, N., Matoussi, A. and Ngoupeyou, A., Quadratic exponential semimartingales and application to BSDEs with jumps, arXiv: 1603.06191, 2016. Google Scholar

[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.  Google Scholar

[11]

He, S., Wang, J. and Yan, J., Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992. Google Scholar

[12]

Jacod, J. and Shiryaev, A. N., Limit Theorems for Stochastic Processes, 2nd ed., Springer, Berlin, 2003. Google Scholar

[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. Google Scholar

[14]

Jeanblanc, M., Matoussi, A. and Ngoupeyou, A., Robust utility maximization problem in a discontinuous filtration, arXiv: 1201.2690v3, 2013. Google Scholar

[15]

Kim, E., Nie, T. and Rutkowski, M., American options in nonlinear markets, Electronic Journal of Probability, 2021, 26: 1−41. Google Scholar

[16]

Kim, E., Nie, T. and Rutkowski, M., Arbitrage-free pricing of game options in nonlinear markets, arXiv: 1807.05448v1, 2018. Google Scholar

[17]

Kusuoka, S., A remark on default risk models, Advances in Mathematical Economics, 1999, 1: 69−82. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

Peng, S. and Xu, X., BSDEs with random default time and their applications to default risk, arXiv: 0910.2091, 2009. Google Scholar

[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. Google Scholar

[26]

Protter, P. E., Stochastic Integration and Differential Equations, 2nd ed., Springer, Berlin, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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