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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

Dmytro Marushkevych and  Alexandre Popier ,

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 inft→T Y (t) = ξ = Y (T). Hence, we extend known results for a non-Markovian terminal condition.
Keywords: Backward stochastic differential equations, Functional stochastic calculus, Singularity.
Mathematics Subject Classification: 60G99;60H99.
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.

[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.

[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.

[4]

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

[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.

[6]

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

[7]

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

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[6]

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

[7]

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

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[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.

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