Given p ∈ (1, 2), we study 𝕃$$-solutions of a reflected backward stochastic differential equation with jumps (RBSDEJ) whose generator g is Lipschitz continuous in (y, z, u). Based on a general comparison theorem as well as the optimal stopping theory for uniformly integrable processes under jump filtration, we show that such a RBSDEJ with p-integrable parameters admits a unique 𝕃$$ solution via a fixed-point argument. The Y -component of the unique 𝕃$$ solution can be viewed as the Snell envelope of the reflecting obstacle 𝔏 under g-evaluations, and the first time Y meets 𝔏 is an optimal stopping time for maximizing the g-evaluation of reward 𝔏.
Keywords: Reflected backward stochastic differential equations with jumps, 𝕃p solutions, comparison theorem, optimal stopping, Snell envelope, Doob–Meyer decomposition, martingale representation theorem, fixed-point argument, $$-evaluations
@article{PS_2020__24_1_935_0,
author = {Yao, Song},
title = {$\mathbb{L}^p$ solutions of reflected backward stochastic differential equations with jumps},
journal = {ESAIM: Probability and Statistics},
pages = {935--962},
year = {2020},
publisher = {EDP Sciences},
volume = {24},
doi = {10.1051/ps/2020026},
mrnumber = {4178791},
zbl = {1454.60084},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2020026/}
}
TY - JOUR
AU - Yao, Song
TI - $\mathbb{L}^p$ solutions of reflected backward stochastic differential equations with jumps
JO - ESAIM: Probability and Statistics
PY - 2020
SP - 935
EP - 962
VL - 24
PB - EDP Sciences
UR - https://www.numdam.org/articles/10.1051/ps/2020026/
DO - 10.1051/ps/2020026
LA - en
ID - PS_2020__24_1_935_0
ER -
%0 Journal Article
%A Yao, Song
%T $\mathbb{L}^p$ solutions of reflected backward stochastic differential equations with jumps
%J ESAIM: Probability and Statistics
%D 2020
%P 935-962
%V 24
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/ps/2020026/
%R 10.1051/ps/2020026
%G en
%F PS_2020__24_1_935_0
Yao, Song. $\mathbb{L}^p$ solutions of reflected backward stochastic differential equations with jumps. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 935-962. doi: 10.1051/ps/2020026
[1] and , Infinite horizon reflected backward SDEs with jumps and RCLL obstacle. Stoch. Anal. Appl. 24 (2006) 1239–1261. | MR | Zbl | DOI
[2] , -solution of reflected generalized BSDEs with non-Lipschitz coefficients. Random Oper. Stoch. Equ. 17 (2009) 201–219. | MR | Zbl | DOI
[3] , Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44 (1973) 384–404. | MR | Zbl | DOI
[4] and , BSDEs with polynomial growth generators. J. Appl. Math. Stoch. Anal. 13 (2000) 207–238. | MR | Zbl | DOI
[5] , , , and , solutions of backward stochastic differential equations. Stoch. Process. Appl. 108 (2003) 109–129. | MR | Zbl | DOI
[6] , and , Reflected BSDEs with nonpositive jumps, and controller-and-stopper games. Stoch. Process. Appl. 125 (2015) 597–633. | MR | Zbl | DOI
[7] and , Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison. Ann. Appl. Probab. 18 (2008) 2041–2069. | MR | Zbl | DOI
[8] , and , Backward stochastic differential equations with constraints on the gains-process Ann. Probab. 26 (1998) 1522–1551. | MR | Zbl
[9] and , Probabilités et potentiel. Hermann, Paris. Chapitres I à IV, Édition entièrement refondue, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. XV, Actualités Scientifiques et Industrielles, No. 1372 (1975). | MR | Zbl
[10] and , Probabilities and Potential. B, Theory of Martingales, Translated from the French by J.P. Wilson. vol. 72 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam (1982). | MR | Zbl
[11] , and , Optimal stopping for dynamic risk measures with jumps and obstacle problems. J. Optim. Theory Appl. 167 (2015) 219–242. | MR | Zbl | DOI
[12] Les aspects probabilistes du contrôle stochastique, in Ninth Saint Flour Probability Summer School—1979 (Saint Flour, 1979), Vol. 876 of Lecture Notes in Mathematics. Springer, Berlin (1981) 73–238. | MR | Zbl
[13] , , , and , Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab. 25 (1997) 702–737. | MR | Zbl | DOI
[14] , , , and , Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab. 25 (1997) 702–737. | MR | Zbl | DOI
[15] , and , Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1–71. | MR | Zbl | DOI
[16] and , Adding constraints to BSDEs with jumps: an alternative to multidimensional reflections. ESAIM: PS 18 (2014) 233–250. | MR | Zbl | Numdam | DOI
[17] , Reflected backward stochastic differential equation with jumps and RCLL obstacle. Bull. Sci. Math. 132 (2008) 690–710. | MR | Zbl | DOI
[18] , , , and , Reflected BSDEs when the obstacle is not right-continuous and optimal stopping. Ann. Appl. Probab. 27 (2017) 3153–3188. | MR | Zbl | DOI
[19] and , Reflected backward stochastic differential equation with jumps and random obstacle. Electron. J. Probab. 8 (2003) 1–20. | MR | Zbl | DOI
[20] and , Reflected backward SDEs with general jumps. Teor. Veroyatnost. i Primenen. 60 (2015) 357–376. | MR | Zbl | DOI
[21] and , -solutions for reflected backward stochastic differential equations. Stoch. Dyn. 12 (2012) 1150016, 35. | MR | Zbl | DOI
[22] and , Limit Theorems for Stochastic Processes, 2nd edn. 2003. Springer-Verlag (1987). | MR | Zbl | DOI
[23] and , Methods of Mathematical Finance, Vol. 39 of Applications of Mathematics (New York). Springer-Verlag, New York (1998). | MR | Zbl
[24] , A sufficient condition for the uniform integrability of exponential martingales. Math. Rep. Toyama Univ. 2 (1979). | MR | Zbl
[25] , Reflected BSDEs with monotone generator. Electron. J. Probab. 17 (2012) 107, 25. | MR | Zbl | DOI
[26] , BSDEs with monotone generator and two irregular reflecting barriers. Bull. Sci. Math. 137 (2013) 268–321. | MR | Zbl | DOI
[27] , Reflected BSDEs on filtered probability spaces. Stoch. Process. Appl. 125 (2015) 4204–4241. | MR | Zbl | DOI
[28] , Discrete-Parameter Martingales, rev. edn. Translated from the French by T.P. Speed, North-Holland Mathematical Library, Vol. 10. North-Holland Publishing Co., Amsterdam (1975). | MR | Zbl
[29] and , Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14 (1990) 55–61. | MR | Zbl | DOI
[30] , Backward SDE and Related g-Expectation, Vol. 364 of Pitman Res. Notes Math. Ser. Longman, Harlow (1997). | MR | Zbl
[31] , Dynamical evaluations. C. R. Math. Acad. Sci. Paris 339 (2004) 585–589. | MR | Zbl | DOI
[32] , Nonlinear Expectations, Nonlinear Evaluations and Risk Measures, Vol. 1856 of Lecture Notes in Math. Springer, Berlin (2004). | MR | Zbl | DOI
[33] , Stochastic Integration and Differential Equations, A new approach. Vol. 21 of Applications of Mathematics (New York). Springer-Verlag, Berlin (1990). | MR | Zbl
[34] and , BSDEs with jumps, optimization and applications to dynamic risk measures. Stoch. Process. Appl. 123 (2013) 3328–3357. | MR | Zbl | DOI
[35] and , Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps. Stoch. Process. Appl. 124 (2014) 3031–3054. | MR | Zbl | DOI
[36] and , Diffusions, Markov Processes and Martingales, Itô calculus, Reprint of the secondedition (1994). Vol. 2 of Cambridge Mathematical Library. Cambridge University Press, Cambridge (2000). | Zbl
[37] and , Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32 (1994) 1447–1475. | MR | Zbl | DOI
[38] , solutions of backward stochastic differential equations with jumps. Stoch. Process. Appl. 127 (2017) 3465–3511. | MR | Zbl | DOI
[39] , On -evaluations with domains under jump filtration. Stoch. Anal. Appl. 36 (2018) 40–102. | MR | Zbl | DOI
Cité par Sources :
