×

zbMATH — the first resource for mathematics

Minimal supersolutions of convex BSDEs. (English) Zbl 1284.60116
Summary: We study the nonlinear operator of mapping the terminal value \(\xi\) to the corresponding minimal supersolution of a backward stochastic differential equation with the generator being monotone in \(y\), convex in \(z\), jointly lower semicontinuous and bounded below by an affine function of the control variable \(z\). We show existence, uniqueness, monotone convergence, Fatou’s lemma and lower semicontinuity of this operator. We provide a comparison principle for minimal supersolutions of BSDEs.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ankirchner, S., Imkeller, P. and Dos Reis, G. (2007). Classical and variational differentiability of BSDEs with quadratic growth. Electron. J. Probab. 12 1418-1453 (electronic). · Zbl 1138.60042
[2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203-228. · Zbl 0980.91042
[3] Barlow, M. T. and Protter, P. (1990). On convergence of semimartingales. In Séminaire de Probabilités , XXIV , 1988 / 89. Lecture Notes in Math. 1426 188-193. Springer, Berlin. · Zbl 0703.60041
[4] Bion-Nadal, J. (2009). Time consistent dynamic risk processes. Stochastic Process. Appl. 119 633-654. · Zbl 1156.91359
[5] Briand, P. and Confortola, F. (2008). BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces. Stochastic Process. Appl. 118 818-838. · Zbl 1136.60337
[6] Briand, P. and Hu, Y. (2008). Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields 141 543-567. · Zbl 1141.60037
[7] Cheridito, P., Delbaen, F. and Kupper, M. (2006). Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11 57-106. · Zbl 1184.91109
[8] Cheridito, P. and Stadje, M. (2012). Existence, minimality and approximation of solutions to BSDEs with convex drivers. Stochastic Process. Appl. 122 1540-1565. · Zbl 1252.60053
[9] Delbaen, F. (2006). The structure of \(m\)-stable sets and in particular of the set of risk neutral measures. In In Memoriam Paul-André Meyer : Séminaire de Probabilités XXXIX. Lecture Notes in Math. 1874 215-258. Springer, Berlin. · Zbl 1121.60043
[10] Delbaen, F., Hu, Y. and Bao, X. (2011). Backward SDEs with superquadratic growth. Probab. Theory Related Fields 150 145-192. · Zbl 1253.60072
[11] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463-520. · Zbl 0865.90014
[12] Delbaen, F. and Schachermayer, W. (1999). A compactness principle for bounded sequences of martingales with applications. In Seminar on Stochastic Analysis , Random Fields and Applications ( Ascona , 1996). Progress in Probability 45 137-173. Birkhäuser, Basel. · Zbl 0939.60024
[13] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B : Theory of Martingales. North-Holland Mathematics Studies 72 . North-Holland, Amsterdam. · Zbl 0494.60002
[14] Dudley, R. M. (1977). Wiener functionals as Itô integrals. Ann. Probab. 5 140-141. · Zbl 0359.60071
[15] Duffie, D. and Epstein, L. G. (1992). Stochastic differential utility. Econometrica 60 353-394. · Zbl 0763.90005
[16] El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S. and Quenez, M. C. (1997). Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab. 25 702-737. · Zbl 0899.60047
[17] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1-71. · Zbl 0884.90035
[18] El Karoui, N. and Quenez, M.-C. (1995). Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33 29-66. · Zbl 0831.90010
[19] Föllmer, H. and Penner, I. (2006). Convex risk measures and the dynamics of their penalty functions. Statist. Decisions 24 61-96. · Zbl 1186.91119
[20] Föllmer, H. and Schied, A. (2002). Convex measures of risk and trading constraints. Finance Stoch. 6 429-447. · Zbl 1041.91039
[21] Föllmer, H. and Schied, A. (2004). Stochastic Finance : An Introduction in Discrete Time , 2 ed. de Gruyter Studies in Mathematics 27 . de Gruyter, Berlin. · Zbl 1126.91028
[22] Gerdes, H., Heyne, G. and Kupper, M. (2012). Stability of minimal supersolutions of BSDEs.
[23] Harrison, J. M. and Pliska, S. R. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Process. Appl. 11 215-260. · Zbl 0482.60097
[24] Heyne, G. (2012). Essays on minimal supersolutions of BSDEs and on cross hedging in incomplete markets. Ph.D. thesis, Humboldt Universität zu Berlin.
[25] Heyne, G., Kupper, M. and Mainberger, C. (2012). Minimal supersolutions of BSDEs with lower semicontinuous generators. Ann. Inst. Henri Poincaré Probab. Stat. · Zbl 1296.60173
[26] Karatzas, I. and Shreve, S. E. (2004). Brownian Motion and Stochastic Calculus ( Graduate Texts in Mathematics ). Springer, New York. · Zbl 0734.60060
[27] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558-602. · Zbl 1044.60045
[28] Maccheroni, F., Marinacci, M. and Rustichini, A. (2006). Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74 1447-1498. · Zbl 1187.91066
[29] Peng, S. (1997). Backward SDE and related \(g\)-expectation. In Backward Stochastic Differential Equations ( Paris , 1995 - 1996). Pitman Res. Notes Math. Ser. 364 141-159. Longman, Harlow. · Zbl 0892.60066
[30] Peng, S. (1999). Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type. Probab. Theory Related Fields 113 473-499. · Zbl 0953.60059
[31] Peng, S. (2007). \(G\)-expectation, \(G\)-Brownian motion and related stochatic calculus of Itô type. Stoch. Anal. Appl. 2 541-567. · Zbl 1131.60057
[32] Peng, S. (2008). Multi-dimensional \(G\)-Brownian motion and related stochastic calculus under \(G\)-expectation. Stochastic Process. Appl. 118 2223-2253. · Zbl 1158.60023
[33] Peng, S. and Xu, M. (2005). The smallest \(g\)-supermartingale and reflected BSDE with single and double \(L^{2}\) obstacles. Ann. Inst. Henri Poincaré Probab. Stat. 41 605-630. · Zbl 1071.60049
[34] Protter, P. E. (2004). Stochastic Integration and Differential Equations : Stochastic Modelling and Applied Probability , 2nd ed. Applications of Mathematics ( New York ) 21 . Springer, Berlin. · Zbl 1041.60005
[35] Réveillac, A. (2011). Weak martingale representation for continuous Markov processes and application to quadratic growth BSDEs.
[36] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Grundlehren der Mathematischen Wissenschaften 293 . Springer, Berlin. · Zbl 0917.60006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.