The Annals of Applied Probability

A dual characterization of self-generation and exponential forward performances

Gordan Žitković

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We propose a mathematical framework for the study of a family of random fields—called forward performances—which arise as numerical representation of certain rational preference relations in mathematical finance. Their spatial structure corresponds to that of utility functions, while the temporal one reflects a Nisio-type semigroup property, referred to as self-generation. In the setting of semimartingale financial markets, we provide a dual formulation of self-generation in addition to the original one, and show equivalence between the two, thus giving a dual characterization of forward performances. Then we focus on random fields with an exponential structure and provide necessary and sufficient conditions for self-generation in that case. Finally, we illustrate our methods in financial markets driven by Itô-processes, where we obtain an explicit parametrization of all exponential forward performances.

Article information

Ann. Appl. Probab., Volume 19, Number 6 (2009), 2176-2210.

First available in Project Euclid: 25 November 2009

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Zentralblatt MATH identifier

Primary: 91B16: Utility theory
Secondary: 91B28

Exponential utility forward performances incomplete markets utility maximization convex duality random fields mathematical finance


Žitković, Gordan. A dual characterization of self-generation and exponential forward performances. Ann. Appl. Probab. 19 (2009), no. 6, 2176--2210. doi:10.1214/09-AAP607.

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  • [1] Acciaio, B. (2005). Absolutely continuous optimal martingale measures. Statist. Decisions 23 81–100.
  • [2] Aliprantis, C. D. and Border, K. C. (1999). Infinite-Dimensional Analysis, 2nd ed. Springer, Berlin.
  • [3] Berrier, F. P. Y. S., Rogers, L. C. G. and Tehranchi, M. R. (2009). A characterization of forward utility functions. Preprint.
  • [4] Bellini, F. and Frittelli, M. (2002). On the existence of minimax martingale measures. Math. Finance 12 1–21.
  • [5] Choulli, T., Stricker, C. and Li, J. (2007). Minimal Hellinger martingale measures of order q. Finance Stoch. 11 399–427.
  • [6] Cox, J. C. and Huang, C.-F. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Econom. Theory 49 33–83.
  • [7] Cvitanić, J., Schachermayer, W. and Wang, H. (2001). Utility maximization in incomplete markets with random endowment. Finance Stoch. 5 259–272.
  • [8] Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M. and Stricker, C. (2002). Exponential hedging and entropic penalties. Math. Finance 12 99–123.
  • [9] Delbaen, F. and Schachermayer, W. (2006). The Mathematics of Arbitrage. Springer, Berlin.
  • [10] Ekeland, I. and Témam, R. (1999). Convex Analysis and Variational Problems, English ed. Classics in Applied Mathematics 28. SIAM, Philadelphia, PA.
  • [11] 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.
  • [12] Föllmer, H. and Kramkov, D. (1997). Optional decompositions under constraints. Probab. Theory Related Fields 109 1–25.
  • [13] Föllmer, H. and Schied, A. (2004). Stochastic Finance, extended ed. de Gruyter Studies in Mathematics 27. de Gruyter, Berlin.
  • [14] Frittelli, M. (2000). The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance 10 39–52.
  • [15] Frittelli, M. and Scandolo, G. (2006). Risk measures and capital requirements for processes. Math. Finance 16 589–612.
  • [16] He, H. and Pearson, N. D. (1991). Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite-dimensional case. J. Econom. Theory 54 259–304.
  • [17] Henderson, V. and Hobson, D. (2007). Horizon-unbiased utility functions. Stochastic Process. Appl. 117 1621–1641.
  • [18] Kabanov, Y. M. and Stricker, C. (2002). On the optimal portfolio for the exponential utility maximization: Remarks to the six-author paper “Exponential hedging and entropic penalties” by F. Delbaen, P. Grandits, T. Rheinländer, D. Samperi, M. Schweizer and C. Stricker. Math. Finance 12 125–134.
  • [19] Karatzas, I., Lehoczky, J. P., Shreve, S. E. and Xu, G.-L. (1990). Optimality conditions for utility maximization in an incomplete market. In Analysis and Optimization of Systems (Antibes, 1990). Lecture Notes in Control and Inform. Sci. 144 3–23. Springer, Berlin.
  • [20] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [21] Karatzas, I. and Žitković, G. (2003). Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31 1821–1858.
  • [22] Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 904–950.
  • [23] Mémin, J. (1980). Espaces de semi martingales et changement de probabilité. Z. Wahrsch. Verw. Gebiete 52 9–39.
  • [24] Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. Rev. Econom. Statist. 51 247–257.
  • [25] Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3 373–413.
  • [26] Musiela, M. and Zariphopoulou, T. (2009). Backward and forward utilities and the associated pricing systems: The case study of the binomial model. Preprint.
  • [27] Musiela, M. and Zariphopoulou, T. (2009). The single period binomial model. In Indifference pricing (R. Carmona, ed.) 3–43. Princeton Univ. Press.
  • [28] Musiela, M. and Zariphopoulou, T. (2006). Investment and valuation under backward and forward dynamic exponential utilities in a stochastic factor model. In Festschrift for Dilip Madan 303–334. Birkhäuser, Boston.
  • [29] Musiela, M. and Zariphopoulou, T. (2008). Optimal asset allocation under forward exponential criteria. In Markov Processes and Related Topics: A Festschrift for T. G. Kurtz 285–300. IMS, Beachwood, OH.
  • [30] Musiela, M. and Zariphopoulou, T. (2009). Investment performance measurement, risk tolerance and optimal portfolio choice. To appear.
  • [31] Nisio, M. (1976/77). On a non-linear semi-group attached to stochastic optimal control. Publ. Res. Inst. Math. Sci. 12 513–537.
  • [32] Owen, M. and Žitković, G. (2009). Optimal investment with an unbounded random endowment and utility-based pricing. Math. Finance 19 129–159.
  • [33] Pliska, S. R. (1986). A stochastic calculus model of continuous trading: Optimal portfolios. Math. Oper. Res. 11 370–382.
  • [34] Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd ed. Applications of Mathematics (New York) 21. Springer, Berlin. Stochastic Modelling and Applied Probability.
  • [35] Rockafellar, R. T. and Wets, R. J.-B. (1998). Variational Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 317. Springer, Berlin.
  • [36] Schachermayer, W. (2001). Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Probab. 11 694–734.
  • [37] Schachermayer, W. (2003). A super-martingale property of the optimal portfolio process. Finance Stoch. 7 433–456.
  • [38] Zariphopoulou, T. and Žitković, G. (2009). Maturity-independent risk measures. To appear.
  • [39] Žitković, G. (2005). Utility maximization with a stochastic clock and an unbounded random endowment. Ann. Appl. Probab. 15 748–777.
  • [40] Žitković, G. (2009). A decision-theoretic foundation of forward performances. To appear.