The Annals of Probability

Minimal supersolutions of convex BSDEs

Samuel Drapeau, Gregor Heyne, and Michael Kupper

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Abstract

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.

Article information

Source
Ann. Probab., Volume 41, Number 6 (2013), 3973-4001.

Dates
First available in Project Euclid: 20 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1384957780

Digital Object Identifier
doi:10.1214/13-AOP834

Mathematical Reviews number (MathSciNet)
MR3161467

Zentralblatt MATH identifier
1284.60116

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]

Keywords
Supersolutions of backward stochastic differential equations nonlinear expectations supermartingales

Citation

Drapeau, Samuel; Heyne, Gregor; Kupper, Michael. Minimal supersolutions of convex BSDEs. Ann. Probab. 41 (2013), no. 6, 3973--4001. doi:10.1214/13-AOP834. https://projecteuclid.org/euclid.aop/1384957780


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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).
  • [2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203–228.
  • [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.
  • [4] Bion-Nadal, J. (2009). Time consistent dynamic risk processes. Stochastic Process. Appl. 119 633–654.
  • [5] Briand, P. and Confortola, F. (2008). BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces. Stochastic Process. Appl. 118 818–838.
  • [6] Briand, P. and Hu, Y. (2008). Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields 141 543–567.
  • [7] Cheridito, P., Delbaen, F. and Kupper, M. (2006). Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11 57–106.
  • [8] Cheridito, P. and Stadje, M. (2012). Existence, minimality and approximation of solutions to BSDEs with convex drivers. Stochastic Process. Appl. 122 1540–1565.
  • [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.
  • [10] Delbaen, F., Hu, Y. and Bao, X. (2011). Backward SDEs with superquadratic growth. Probab. Theory Related Fields 150 145–192.
  • [11] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463–520.
  • [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.
  • [13] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B: Theory of Martingales. North-Holland Mathematics Studies 72. North-Holland, Amsterdam.
  • [14] Dudley, R. M. (1977). Wiener functionals as Itô integrals. Ann. Probab. 5 140–141.
  • [15] Duffie, D. and Epstein, L. G. (1992). Stochastic differential utility. Econometrica 60 353–394.
  • [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.
  • [17] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
  • [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.
  • [19] Föllmer, H. and Penner, I. (2006). Convex risk measures and the dynamics of their penalty functions. Statist. Decisions 24 61–96.
  • [20] Föllmer, H. and Schied, A. (2002). Convex measures of risk and trading constraints. Finance Stoch. 6 429–447.
  • [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.
  • [22] Gerdes, H., Heyne, G. and Kupper, M. (2012). Stability of minimal supersolutions of BSDEs. Preprint.
  • [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.
  • [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. To appear.
  • [26] Karatzas, I. and Shreve, S. E. (2004). Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics). Springer, New York.
  • [27] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558–602.
  • [28] Maccheroni, F., Marinacci, M. and Rustichini, A. (2006). Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74 1447–1498.
  • [29] Peng, S. (1997). Backward SDE and related $g$-expectation. In Backward Stochastic Differential Equations (Paris, 19951996). Pitman Res. Notes Math. Ser. 364 141–159. Longman, Harlow.
  • [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.
  • [31] Peng, S. (2007). $G$-expectation, $G$-Brownian motion and related stochatic calculus of Itô type. Stoch. Anal. Appl. 2 541–567.
  • [32] Peng, S. (2008). Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation. Stochastic Process. Appl. 118 2223–2253.
  • [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.
  • [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.
  • [35] Réveillac, A. (2011). Weak martingale representation for continuous Markov processes and application to quadratic growth BSDEs. Preprint.
  • [36] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin.