Limit behaviour of BSDE with jumps and with singular terminal condition

Popier, A.

ESAIM: Probability and Statistics, Tome 20 (2016), pp. 480-509.

We study the behaviour at the terminal time $T$ of the minimal solution of a backward stochastic differential equation when the terminal data can take the value $+\infty$ with positive probability. In a previous paper [T. Kruse and A. Popier, Stoch. Process. Appl. 126 (2016) 2554–2592], we have proved existence of this minimal solution (in a weak sense) in a quite general setting. But two questions arise in this context and were still open: is the solution right continuous with left limits on $[0,T]$? In other words does the solution have a left limit at time $T$? The second question is: is this limit equal to the terminal condition? In this paper, under additional conditions on the generator and the terminal condition, we give a positive answer to these two questions.

Classification : 60G99, 60H99, 60J15
Mots clés : Backward stochastic differential equations, jumps, general filtration, singularity

A. Aksamit, T. Choulli and M. Jeanblanc, Classification of random times and applications. Preprint (2016). | arXiv

S. Ankirchner, M. Jeanblanc and T. Kruse, BSDEs with Singular Terminal Condition and a Control Problem with Constraints. SIAM J. Control Optim. 52 (2014) 893–913. | DOI | Zbl

G. Barles, R. Buckdahn and É. Pardoux, Backward stochastic differential equations and integral-partial differential equations. Stochastics Stochastics Rep. 60 (1997) 57–83. | DOI | Zbl

J.-M. Bismut, Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44 (1973) 384–404. | DOI | Zbl

B. Bouchard, D. Possamaï, X. Tan and C. Zhou, A unified approach to a priori estimates for supersolutions of BSDEs in general filtrations. Preprint (2015). | arXiv

Ł. Delong, Backward stochastic differential equations with jumps and their actuarial and financial applications. BSDEs with jumps. Eur. Actuarial Acad. (EAA) Ser. Springer, London (2013). | Zbl

D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics. Reprint of the 1998 edition. Springer Verlag, Berlin (2001). | Zbl

P. Graewe, U. Horst and J. Qiu, A Non-Markovian Liquidation Problem and Backward SPDEs with Singular Terminal Conditions. SIAM J. Control. Optim. 53 (2015) 690–711. | DOI | MR | Zbl

J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes, Vol. 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2nd edition (2003). | MR | Zbl

J. Jacod and A.V. Skorohod, Jumping filtrations and martingales with finite variation. In Séminaire de Probabilités, XXVIII. Vol. 1583 of Lect. Notes Math. Springer, Berlin (1994) 21–35. | Numdam | MR | Zbl

M. Jeanblanc, T. Mastrolia, D. Possamaï and A. Réveillac, Utility maximization with random horizon: a BSDE approach. Int. J. Theor. Appl. Finance 18 (2015) 1550045. | DOI | MR | Zbl

M. Jeanblanc and A. Réveillac, A note on BSDEs with singular driver coefficients. In Arbitrage, credit and informational risks, Vol. 5 of Peking Univ. Ser. Math. World Sci. Publ., Hackensack, NJ (2014) 207–224. | MR

O. Kallenberg, Foundations of modern probability. Springer, Berlin (2002). | MR | Zbl

T. Kruse and A. Popier, BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration. Stochastics 88 (2016) 491–539. | DOI | MR | Zbl

T. Kruse and A. Popier, Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting. Stoch. Process. Appl. 126 (2016) 2554–2592. | DOI | MR | Zbl

D. Nualart, The Malliavin calculus and related topics. Probability and its Applications. New York. Springer-Verlag, Berlin, 2nd edition (2006). | MR

B. Øksendal and A. Sulem, Applied stochastic control of jump diffusions. Universitext. Springer, Berlin, 2nd edition (2007). | MR

É. Pardoux and S.G. Peng, Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 (1990) 55–61. | DOI | MR | Zbl

E. Pardoux and A. Rascanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Vol. 69 of Stochastic Modelling and Applied Probability. Springer-Verlag (2014). | MR | Zbl

A. Popier, Backward stochastic differential equations with singular terminal condition. Stochastic Process. Appl. 116 (2006) 2014–2056. | DOI | MR | Zbl

Ph. E. Protter, Stochastic integration and differential equations. Applications of Mathematics (New York). Vol. 21 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2nd edition (2004). | MR | Zbl

M.-C. Quenez and A. Sulem, BSDEs with jumps, optimization and applications to dynamic risk measures. Stochastic Process. Appl. 123 (2013) 3328–3357. | DOI | MR | Zbl

M. Royer, Backward stochastic differential equations with jumps and related non-linear expectations. Stochastic Process. Appl. 116 (2006) 1358–1376. | DOI | MR | Zbl