Convex Duality in Stochastic Optimization and Mathematical Finance

Published Online:https://doi.org/10.1287/moor.1110.0485

References

  • Attouch H., Buttazzo G., Michaille G.Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization (2005) 6(Society for Industrial and Applied Mathematics (SIAM), Philadelphia) MPS/SIAM Series on OptimizationGoogle Scholar
  • Back K., Pliska S. R. The shadow price of information in continuous time decision problems. Stochastics (1987) 22(2):151–186CrossrefGoogle Scholar
  • Balder E. J. Infinite-dimensional extension of a theorem of Komlós. Probab. Theory Related Fields (1989) 81(2):185–188CrossrefGoogle Scholar
  • Beale E. M. L. On minimizing a convex function subject to linear inequalities. J. Roy. Statist. Soc. Ser. B. (1955) 17(2):173–184Google Scholar
  • Bertsekas D. P. Necessary and sufficient conditions for existence of an optimal portfolio. J. Econom. Theory (1974) 8(2):235–247CrossrefGoogle Scholar
  • Biagini S., Cont R. Expected utility maximization: The dual approach. Encyclopedia of Quantitative Finance (2010) (Wiley, New York) . ForthcomingCrossrefGoogle Scholar
  • Biagini S., Frittelli M. A unified framework for utility maximization problems: An Orlicz spaces approach. Ann. Appl. Probab. (2008) 18(3):929–966CrossrefGoogle Scholar
  • Castaing C., Valadier M.Convex Analysis and Measurable Multifunctions (1977) 580(Springer-Verlag, Berlin) CrossrefGoogle Scholar
  • Choirat C., Hess C., Seri R. A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach. Ann. Probab. (2003) 31(1):63–92CrossrefGoogle Scholar
  • Cvitanić J., Karatzas I. Convex duality in constrained portfolio optimization. Ann. Appl. Probab. (1992) 2(4):767–818CrossrefGoogle Scholar
  • Dalang R. C., Morton A., Willinger W. Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stochastics Stochastics Rep. (1990) 29(2):185–201CrossrefGoogle Scholar
  • Dantzig G. B. Linear programming under uncertainty. Management Sci. (1955) 1(3–4):197–206LinkGoogle Scholar
  • Davis M. H. A. Dynamic optimization: A grand unification. Proc. 31st IEEE Conf. Decision and Control (1994) 2Tucson, AZ:2035–2036Google Scholar
  • Davis M. H. A., Burstein G. A deterministic approach to stochastic optimal control with application to anticipative control. Stochastics Stochastics Rep. (1992) 40(3&4):203–256CrossrefGoogle Scholar
  • Davis M. H. A., Karatzas I., Kelly F. A deterministic approach to optimal stopping. Probability, Statistics and Optimisation: A Tribute to Peter Whittle (1994) (Wiley, Chichester, UK) 455–466Google Scholar
  • Delbaen F., Schachermayer W.The Mathematics of Arbitrage (2006) (Springer Finance, Springer-Verlag, Berlin/Heidelberg) Google Scholar
  • Dempster M. A. H., Evstigneev I. V., Taksar M. I. Asset pricing and hedging in financial markets with transaction costs: An approach based on the von Neumann-Gale model. Ann. Finance (2006) 2(4):327–355CrossrefGoogle Scholar
  • Ekeland I., Temam R.Convex Analysis and Variational Problems (1976) (North-Holland Publishing Co., Amsterdam) . [Translated from the French, Studies in Mathematics and Its Applications, Vol. 1.]Google Scholar
  • Föllmer H., Schied A.Stochastic Finance. de Gruyter Studies in Mathematics (2004) 27(Walter de Gruyter & Co., Berlin) . [Extended ed. Stochastic Finance, An Introduction in Discrete Time.]CrossrefGoogle Scholar
  • Grothendieck A.Topological Vector Spaces (1973) (Gordon and Breach Science Publishers, New York) . [Translated from the French by Orlando Chaljub, Notes on Mathematics and Its Applications.]Google Scholar
  • Haugh M. B., Kogan L. Pricing American options: A duality approach. Oper. Res. (2004) 52(2):258–270LinkGoogle Scholar
  • Hiai F., Umegaki H. Integrals, conditional expectations, and martingales of multivalued functions. J. Multivariate Anal. (1977) 7(1):149–182CrossrefGoogle Scholar
  • Jouini E., Napp C., Schachermayer W. Arbitrage and state price deflators in a general intertemporal framework. J. Math. Econom. (2005) 41(6):722–734CrossrefGoogle Scholar
  • Kabanov Y. M. Hedging and liquidation under transaction costs in currency markets. Finance Stochastics (1999) 3(2):237–248CrossrefGoogle Scholar
  • Kabanov Y. M., Safarian M. Mathematical theory. Markets with Transaction Costs (2009) (Springer Finance, Springer-Verlag, Berlin) Google Scholar
  • Kabanov Y. M., Stricker C. A teachers' note on no-arbitrage criteria. Séminaire de Probabilités, XXXV (2001) 1755(Springer, Berlin) 149–152CrossrefGoogle Scholar
  • Karatzas I., Žitković G. Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. (2003) 31(4):1821–1858CrossrefGoogle Scholar
  • King A. J. Duality and martingales: A stochastic programming perspective on contingent claims. Math. Programming Ser. B (2002) 91(3):543–562CrossrefGoogle Scholar
  • Klein I., Rogers L. C. G. Duality in optimal investment and consumption problems with market frictions. Math. Finance (2007) 17(2):225–247CrossrefGoogle Scholar
  • Komlós J. A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungary (1967) 18:217–229CrossrefGoogle Scholar
  • Korf L. A. Stochastic programming duality: ℒ∞ multipliers for unbounded constraints with an application to mathematical finance. Math. Programming Ser. A (2004) 99(2):241–259CrossrefGoogle Scholar
  • Kramkov D., Schachermayer W. The condition on the asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. (1999) 9(3):904–950CrossrefGoogle Scholar
  • Kreher D. Hedging of portfolio-valued claims under convex transaction costs and portfolio constraints. (2009) . Master's thesis, Humboldt-Universität zu Berlin, BerlinGoogle Scholar
  • Pennanen T. Arbitrage and deflators in illiquid markets. Finance Stochastics (2011) 15(1):57–83CrossrefGoogle Scholar
  • Pennanen T. Superhedging in illiquid markets. Math. Finance (2011) . ForthcomingCrossrefGoogle Scholar
  • Pennanen T., Penner I. Hedging of claims with physical delivery under convex transaction costs. SIAM J. Financial Math. (2010) 1:158–178CrossrefGoogle Scholar
  • Pliska S. R.Introduction to Mathematical Finance: Discrete Time Models (1997) (Blackwell, Malden, MA) Google Scholar
  • Rockafellar R. T. Integrals which are convex functionals. Pacific J. Math. (1968) 24(3):525–539CrossrefGoogle Scholar
  • Rockafellar R. T.Convex Analysis (1970) 28(Princeton University Press, Princeton, NJ) Princeton Mathematical SeriesCrossrefGoogle Scholar
  • Rockafellar R. T. Integrals which are convex functionals. II. Pacific J. Math. (1971) 39(2):439–469CrossrefGoogle Scholar
  • Rockafellar R. T.Conjugate Duality and Optimization (1974) (Society for Industrial and Applied Mathematics, Philadelphia) CrossrefGoogle Scholar
  • Rockafellar R. T. Integral functionals, normal integrands and measurable selections. Nonlinear Operators and the Calculus of Variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975) (1976) 543(Springer, Berlin) 157–207CrossrefGoogle Scholar
  • Rockafellar R. T., Wets R. J.-B. Nonanticipativity and L1-martingales in stochastic optimization problems. Math. Programming Stud. (1976) 6:170–187CrossrefGoogle Scholar
  • Rockafellar R. T., Wets R. J.-B. Measures as Lagrange multipliers in multistage stochastic programming. J. Math. Anal. Appl. (1977) 60(2):301–313CrossrefGoogle Scholar
  • Rockafellar R. T., Wets R. J.-B. The optimal recourse problem in discrete time: L1-multipliers for inequality constraints. SIAM J. Control Optim. (1978) 16(1):16–36CrossrefGoogle Scholar
  • Rockafellar R. T., Wets R. J.-B. Deterministic and stochastic optimization problems of Bolza type in discrete time. Stochastics (1983) 10(3–4):273–312CrossrefGoogle Scholar
  • Rockafellar R. T., Wets R. J.-B.Variational Analysis (1998) 317(Springer-Verlag, Berlin) Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]CrossrefGoogle Scholar
  • Rogers L. C. G. Monte Carlo valuation of American options. Math. Finance (2002) 12(3):271–286CrossrefGoogle Scholar
  • Rokhlin D. B. The Kreps-Yan theorem for L∞. Int. J. Math. Math. Sci. (2005) 2005(17):2749–2756CrossrefGoogle Scholar
  • Schachermayer W. A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insurance Math. Econom. (1992) 11(4):249–257CrossrefGoogle Scholar
  • Schachermayer W. The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance (2004) 14(1):19–48CrossrefGoogle Scholar
  • Shapiro A., Dentcheva D., Ruszczyński A.Lectures on Stochastic Programming: Modeling and Theory (2009) 9(SIAM, Philadelphia) MPS/SIAM Series on OptimizationCrossrefGoogle Scholar
  • Shiryaev A. N.Probability. Graduate Texts in Mathematics (1996) 952nd ed.(Springer-Verlag, New York) . [Translated from the first (1980) Russian edition by R. P. Boas.]Google Scholar
  • Tian S., Wets R. J.-B. Pricing contingent claims: A computational compatible approach. (2006) . Working paper, University of California at Davis, DavisGoogle Scholar
  • Wets R. J.-B., Bensoussan A., Lions J. L. On the relation between stochastic and deterministic optimization. Control Theory, Numerical Methods and Computer Systems Modelling (1975) 107(Springer, New York) 350–361CrossrefGoogle Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.