January  2020, 5: 1 doi: 10.1186/s41546-020-0043-5

Limit behaviour of the minimal solution of a BSDE with singular terminal condition in the non Markovian setting

Laboratoire Manceau de Mathématiques, Le Mans Université, Avenue Olivier Messiaen, 72085 Le Mans cedex 9, France

Received  March 08, 2019 Published  February 2020

We use the functional Itô calculus to prove that the solution of a BSDE with singular terminal condition verifies at the terminal time: lim inftT Y (t) = ξ = Y (T). Hence, we extend known results for a non-Markovian terminal condition.
Citation: Dmytro Marushkevych, Alexandre Popier. Limit behaviour of the minimal solution of a BSDE with singular terminal condition in the non Markovian setting. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 1-. doi: 10.1186/s41546-020-0043-5
References:
[1]

Ankirchner, S., M. Jeanblanc, and T. Kruse. (2014). BSDEs with Singular Terminal Condition and a Control Problem with Constraints, SIAM J. Control Optim. 52, no. 2, 893–913. Google Scholar

[2]

Bank, P. and M. Voß. (2018). Linear quadratic stochastic control problems with stochastic terminal constraint, SIAM J. Control Optim. 56, no. 2, 672–699. Google Scholar

[3]

Bouchard, B., D. Possamaï, X. Tan, and C. Zhou. (2018). A unified approach to a priori estimates for supersolutions of BSDEs in general filtrations, Ann. Inst. Henri Poincaré, Probab. Stat. 54, no. 1, 154–172. Google Scholar

[4]

Cont, R. (2016). Functional Itô calculus and functional Kolmogorov equations, Birkhäuser/Springer, CRM Barcelona. Google Scholar

[5]

Cont, R. and D.-A. Fournié. (2010). A functional extension of the Ito formula, C. R. Math. Acad. Sci. Paris. 348, no. 1–2, 57–61. Google Scholar

[6]

Cont, R. and D.-A. Fournié. (2013). Functional Itô calculus and stochastic integral representation of martingales, Ann. Probab. 41, no. 1, 109–133. Google Scholar

[7]

Cont, R. and Y. Lu. (2016). Weak approximation of martingale representations, Stoch. Process. Appl. 126, no. 3, 857–882. Google Scholar

[8]

Dellacherie, C. and P.-A. Meyer. (1980). Probabilités et potentiel. Théorie des martingales, Chapitres V à VIII, Hermann. Google Scholar

[9]

Delong, Ł. (2013). Backward stochastic differential equations with jumps and their actuarial and financial applications, European Actuarial Academy (EAA) Series, Springer, London. BSDEs with jumps. Google Scholar

[10]

Dupire, B. (2009). Functional Itô calculus, Bloomberg Portfolio Research Paper No 2009-04-FRONTIERS. Google Scholar

[11]

Graewe, P., U. Horst, and J. Qiu. (2015). A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions, SIAM J. Control Optim. 53, no. 2, 690–711. Google Scholar

[12]

Kruse, T. and A. Popier. (2016). BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration, Stochastics 88, no. 4, 491–539. Google Scholar

[13]

Kruse, T. and A. Popier. (2016). Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting, Stoch. Process. Appl. 126, no. 9, 2554–2592. Google Scholar

[14]

Kruse, T. and A. Popier. (2017). Lp-solution for BSDEs with jumps in the case p < 2: corrections to the paper ‘BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration, Stochastics 89, no. 8, 1201–1227. Google Scholar

[15]

Pardoux, E. and A. Rascanu. (2014). Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, volume 69 of Stochastic Modelling and Applied Probability, Springer-Verlag. https://doi.org/10.1007/978-3-319-05714-9. Google Scholar

[16]

Popier, A. (2006). Backward stochastic differential equations with singular terminal condition, Stoch.Process. Appl 116, no. 12, 2014–2056. Google Scholar

[17]

Popier, A. (2016). Limit behaviour of bsde with jumps and with singular terminal condition, ESAIM: PS 20, 480–509. Google Scholar

[18]

Protter, P.E. (2004). Stochastic integration and differential equations, volume 21 of Applications of Mathematics (New York), second edition, Springer-Verlag, Berlin. Stochastic Modelling and Applied Probability. Google Scholar

[19]

Quenez, M.-C. and A. Sulem. (2013). BSDEs with jumps, optimization and applications to dynamic risk measures, Stoch. Process. Appl. 123, no. 8, 3328–3357. Google Scholar

[20]

Sezer, A.D., T. Kruse, and A. Popier. (2019). Backward stochastic differential equations with nonMarkovian singular terminal values, Stoch. Dyn. 19, no. 2, 1950006. Google Scholar

show all references

References:
[1]

Ankirchner, S., M. Jeanblanc, and T. Kruse. (2014). BSDEs with Singular Terminal Condition and a Control Problem with Constraints, SIAM J. Control Optim. 52, no. 2, 893–913. Google Scholar

[2]

Bank, P. and M. Voß. (2018). Linear quadratic stochastic control problems with stochastic terminal constraint, SIAM J. Control Optim. 56, no. 2, 672–699. Google Scholar

[3]

Bouchard, B., D. Possamaï, X. Tan, and C. Zhou. (2018). A unified approach to a priori estimates for supersolutions of BSDEs in general filtrations, Ann. Inst. Henri Poincaré, Probab. Stat. 54, no. 1, 154–172. Google Scholar

[4]

Cont, R. (2016). Functional Itô calculus and functional Kolmogorov equations, Birkhäuser/Springer, CRM Barcelona. Google Scholar

[5]

Cont, R. and D.-A. Fournié. (2010). A functional extension of the Ito formula, C. R. Math. Acad. Sci. Paris. 348, no. 1–2, 57–61. Google Scholar

[6]

Cont, R. and D.-A. Fournié. (2013). Functional Itô calculus and stochastic integral representation of martingales, Ann. Probab. 41, no. 1, 109–133. Google Scholar

[7]

Cont, R. and Y. Lu. (2016). Weak approximation of martingale representations, Stoch. Process. Appl. 126, no. 3, 857–882. Google Scholar

[8]

Dellacherie, C. and P.-A. Meyer. (1980). Probabilités et potentiel. Théorie des martingales, Chapitres V à VIII, Hermann. Google Scholar

[9]

Delong, Ł. (2013). Backward stochastic differential equations with jumps and their actuarial and financial applications, European Actuarial Academy (EAA) Series, Springer, London. BSDEs with jumps. Google Scholar

[10]

Dupire, B. (2009). Functional Itô calculus, Bloomberg Portfolio Research Paper No 2009-04-FRONTIERS. Google Scholar

[11]

Graewe, P., U. Horst, and J. Qiu. (2015). A non-Markovian liquidation problem and backward SPDEs with singular terminal conditions, SIAM J. Control Optim. 53, no. 2, 690–711. Google Scholar

[12]

Kruse, T. and A. Popier. (2016). BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration, Stochastics 88, no. 4, 491–539. Google Scholar

[13]

Kruse, T. and A. Popier. (2016). Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting, Stoch. Process. Appl. 126, no. 9, 2554–2592. Google Scholar

[14]

Kruse, T. and A. Popier. (2017). Lp-solution for BSDEs with jumps in the case p < 2: corrections to the paper ‘BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration, Stochastics 89, no. 8, 1201–1227. Google Scholar

[15]

Pardoux, E. and A. Rascanu. (2014). Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, volume 69 of Stochastic Modelling and Applied Probability, Springer-Verlag. https://doi.org/10.1007/978-3-319-05714-9. Google Scholar

[16]

Popier, A. (2006). Backward stochastic differential equations with singular terminal condition, Stoch.Process. Appl 116, no. 12, 2014–2056. Google Scholar

[17]

Popier, A. (2016). Limit behaviour of bsde with jumps and with singular terminal condition, ESAIM: PS 20, 480–509. Google Scholar

[18]

Protter, P.E. (2004). Stochastic integration and differential equations, volume 21 of Applications of Mathematics (New York), second edition, Springer-Verlag, Berlin. Stochastic Modelling and Applied Probability. Google Scholar

[19]

Quenez, M.-C. and A. Sulem. (2013). BSDEs with jumps, optimization and applications to dynamic risk measures, Stoch. Process. Appl. 123, no. 8, 3328–3357. Google Scholar

[20]

Sezer, A.D., T. Kruse, and A. Popier. (2019). Backward stochastic differential equations with nonMarkovian singular terminal values, Stoch. Dyn. 19, no. 2, 1950006. Google Scholar

[1]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075

[2]

Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803

[3]

Joscha Diehl, Jianfeng Zhang. Backward stochastic differential equations with Young drift. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 5-. doi: 10.1186/s41546-017-0016-5

[4]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems & Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002

[5]

Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115

[6]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure &amp; Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103

[7]

Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447

[8]

Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285

[9]

Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565

[10]

Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control & Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013

[11]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[12]

Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905

[13]

Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585

[14]

Kai Liu. On regularity of stochastic convolutions of functional linear differential equations with memory. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1279-1298. doi: 10.3934/dcdsb.2019220

[15]

Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005

[16]

Minghui Song, Liangjian Hu, Xuerong Mao, Liguo Zhang. Khasminskii-type theorems for stochastic functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1697-1714. doi: 10.3934/dcdsb.2013.18.1697

[17]

Ishak Alia. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control & Related Fields, 2020, 10 (4) : 785-826. doi: 10.3934/mcrf.2020020

[18]

Ying Liu, Yabing Sun, Weidong Zhao. Explicit multistep stochastic characteristic approximation methods for forward backward stochastic differential equations. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021044

[19]

Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051

[20]

Feng Bao, Yanzhao Cao, Weidong Zhao. A first order semi-discrete algorithm for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1297-1313. doi: 10.3934/dcdsb.2015.20.1297

 Impact Factor: 

Metrics

  • PDF downloads (27)
  • HTML views (307)
  • Cited by (0)

Other articles
by authors

[Back to Top]