The Annals of Applied Probability

Dual formulation of second order target problems

H. Mete Soner, Nizar Touzi, and Jianfeng Zhang

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Abstract

This paper provides a new formulation of second order stochastic target problems introduced in [SIAM J. Control Optim. 48 (2009) 2344–2365] by modifying the reference probability so as to allow for different scales. This new ingredient enables us to prove a dual formulation of the target problem as the supremum of the solutions of standard backward stochastic differential equations. In particular, in the Markov case, the dual problem is known to be connected to a fully nonlinear, parabolic partial differential equation and this connection can be viewed as a stochastic representation for all nonlinear, scalar, second order, parabolic equations with a convex Hessian dependence.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 1 (2013), 308-347.

Dates
First available in Project Euclid: 25 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1359124388

Digital Object Identifier
doi:10.1214/12-AAP844

Mathematical Reviews number (MathSciNet)
MR3059237

Zentralblatt MATH identifier
1293.60063

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
Stochastic target problem mutually singular probability measures backward SDEs duality

Citation

Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Dual formulation of second order target problems. Ann. Appl. Probab. 23 (2013), no. 1, 308--347. doi:10.1214/12-AAP844. https://projecteuclid.org/euclid.aoap/1359124388


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References

  • [1] Bank, P. and Baum, D. (2004). Hedging and portfolio optimization in financial markets with a large trader. Math. Finance 14 1–18.
  • [2] Bertsekas, D. P. and Shreve, S. E. (1978). Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering 139. Academic Press, New York.
  • [3] Çetin, U., Soner, H. M. and Touzi, N. (2010). Option hedging for small investors under liquidity costs. Finance Stoch. 14 317–341.
  • [4] Chen, Z. and Peng, S. (2000). A general downcrossing inequality for $g$-martingales. Statist. Probab. Lett. 46 169–175.
  • [5] Cheridito, P., Soner, H. M. and Touzi, N. (2005). The multi-dimensional super-replication problem under gamma constraints. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 633–666.
  • [6] Cheridito, P., Soner, H. M., Touzi, N. and Victoir, N. (2007). Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math. 60 1081–1110.
  • [7] Denis, L., Hu, M. and Peng, S. (2011). Function spaces and capacity related to a sublinear expectation: Application to $G$-Brownian motion paths. Potential Anal. 34 139–161.
  • [8] Denis, L. and Martini, C. (2006). A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab. 16 827–852.
  • [9] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
  • [10] 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.
  • [11] Föllmer, H. (1981). Calcul d’Itô sans probabilités. In Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French). Lecture Notes in Math. 850 143–150. Springer, Berlin.
  • [12] Karandikar, R. L. (1995). On pathwise stochastic integration. Stochastic Process. Appl. 57 11–18.
  • [13] Lepeltier, J. P. and Xu, M. (2005). Penalization method for reflected backward stochastic differential equations with one r.c.l.l. barrier. Statist. Probab. Lett. 75 58–66.
  • [14] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control. Lett. 14 55–61.
  • [15] Peng, S. (1999). Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type. Probab. Theory Related Fields 113 473–499.
  • [16] Peng, S. (2004). Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Math. Appl. Sin. Engl. Ser. 20 191–214.
  • [17] Peng, S. (2007). G-Brownian motion and dynamic risk measure under volatility uncertainty. Available at arXiv:0711.2834v1.
  • [18] Soner, H. M. and Touzi, N. (2000). Super-replication under Gamma constraint. SIAM J. Control Optim. 39 73–96.
  • [19] Soner, H. M. and Touzi, N. (2007). Hedging under gamma constraints by optimal stopping and face-lifting. Math. Finance 17 59–79.
  • [20] Soner, H. M. and Touzi, N. (2009). The dynamic programming equation for second order stochastic target problems. SIAM J. Control Optim. 48 2344–2365.
  • [21] Soner, H. M., Touzi, N. and Zhang, J. (2012). Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153 149–190.
  • [22] Soner, H. M., Touzi, N. and Zhang, J. (2011). Quasi-sure stochastic analysis through aggregation. Electron. J. Probab. 16 1844–1879.
  • [23] Soner, H. M., Touzi, N. and Zhang, J. (2011). Martingale representation theorem for the $G$-expectation. Stochastic Process. Appl. 121 265–287.
  • [24] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Springer, Berlin.