Skip to main content

2011 | OriginalPaper | Buchkapitel

Robust utility maximization with unbounded random endowment

verfasst von : Keita Owari

Erschienen in: Advances in Mathematical Economics

Verlag: Springer Japan

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

This paper studies the problem of robust utility maximization with random endowment. When the endowment is possibly unbounded, but satisfies certain integrability conditions, we first prove the fundamental duality relation between the utility maximization and the dual problem, and the existence of a solution to the dual problem. Then the existence of an optimal strategy in a certain choice of admissible class is discussed. As an application, we introduce a robust version of utility indifference prices.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006) Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)
2.
Zurück zum Zitat Ansel, J.-P., Stricker, C.: Couverture des actifs contingents et prix maximum. Ann. Inst. Henri Poincaré 30, 303–315 (1994) Ansel, J.-P., Stricker, C.: Couverture des actifs contingents et prix maximum. Ann. Inst. Henri Poincaré 30, 303–315 (1994)
3.
Zurück zum Zitat Becherer, D.: Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insur. Math. Econ. 33, 1–28 (2003)CrossRef Becherer, D.: Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insur. Math. Econ. 33, 1–28 (2003)CrossRef
4.
Zurück zum Zitat Bellini, F., Frittelli, M.: On the existence of minimax martingale measures. Math. Finance 12, 1–21 (2002)CrossRef Bellini, F., Frittelli, M.: On the existence of minimax martingale measures. Math. Finance 12, 1–21 (2002)CrossRef
5.
Zurück zum Zitat Biagini, S., Frittelli, M.: On the extension of the Namioka–Klee theorem and on the Fatou property for risk measures. In: Delbaen, F., Rasonyi, M., Stricker, C. (eds.) Optimality and Risk: Modern Trends in Mathematical Finance. The Kabanov Festschrift, pp. 1–29. Springer, Berlin (2009)CrossRef Biagini, S., Frittelli, M.: On the extension of the Namioka–Klee theorem and on the Fatou property for risk measures. In: Delbaen, F., Rasonyi, M., Stricker, C. (eds.) Optimality and Risk: Modern Trends in Mathematical Finance. The Kabanov Festschrift, pp. 1–29. Springer, Berlin (2009)CrossRef
6.
Zurück zum Zitat Biagini, S., Frittelli, M., Grasselli, M.: Indifference price with general semimartingales. Math. Finance (2010, forthcoming) Biagini, S., Frittelli, M., Grasselli, M.: Indifference price with general semimartingales. Math. Finance (2010, forthcoming)
7.
Zurück zum Zitat Delbaen, F.: The structure of m-stable sets in particular of the set of risk neutral measures. In: In Memoriam Paul-Andé Meyer, Séminaire de Probabilités XXXIX. Lecture Notes in Mathematics, vol. 1874, pp. 215–258. Springer, Berlin (2006) Delbaen, F.: The structure of m-stable sets in particular of the set of risk neutral measures. In: In Memoriam Paul-Andé Meyer, Séminaire de Probabilités XXXIX. Lecture Notes in Mathematics, vol. 1874, pp. 215–258. Springer, Berlin (2006)
8.
Zurück zum Zitat Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994)CrossRef Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994)CrossRef
9.
Zurück zum Zitat Delbaen, F., Schachermayer, W.: A compactness principle for bounded sequences of martingales with applications. In: Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1996). Progr. Probab., vol. 45, pp. 137–173. Birkhäuser, Basel (1999) Delbaen, F., Schachermayer, W.: A compactness principle for bounded sequences of martingales with applications. In: Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1996). Progr. Probab., vol. 45, pp. 137–173. Birkhäuser, Basel (1999)
10.
Zurück zum Zitat Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M., Stricker, C.: Exponential hedging and entropic penalties. Math. Finance 12, 99–123 (2002)CrossRef Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M., Stricker, C.: Exponential hedging and entropic penalties. Math. Finance 12, 99–123 (2002)CrossRef
11.
Zurück zum Zitat Dellacherie, C., Meyer, P.-A.: Probabilities and Potential A. North-Holland Math. Stud., vol. 29. North-Holland, Amsterdam (1978) Dellacherie, C., Meyer, P.-A.: Probabilities and Potential A. North-Holland Math. Stud., vol. 29. North-Holland, Amsterdam (1978)
12.
Zurück zum Zitat Föllmer, H., Gundel, A.: Robust projections in the class of martingale measures. Illinois J. Math. 50, 439–472 (2006) Föllmer, H., Gundel, A.: Robust projections in the class of martingale measures. Illinois J. Math. 50, 439–472 (2006)
13.
Zurück zum Zitat Föllmer, H., Schied, A., Weber, S.: Robust preferences and robust portfolio choice. In: Bensoussan, A., Zhang, Q., Ciarlet, P.G. (eds.) Mathematical Modelling and Numerical Methods in Finance. Handbook of Numerical Analysis, vol. 15, pp. 29–88. North-Holland, Amsterdam (2009) Föllmer, H., Schied, A., Weber, S.: Robust preferences and robust portfolio choice. In: Bensoussan, A., Zhang, Q., Ciarlet, P.G. (eds.) Mathematical Modelling and Numerical Methods in Finance. Handbook of Numerical Analysis, vol. 15, pp. 29–88. North-Holland, Amsterdam (2009)
14.
Zurück zum Zitat Frittelli, M., Rosazza Gianin, E.: Equivalent formulations of reasonable asymptotic elasticity. Tech. Rep. 12, Dept. Matematica per le Decisioni, University of Florence (2004) Frittelli, M., Rosazza Gianin, E.: Equivalent formulations of reasonable asymptotic elasticity. Tech. Rep. 12, Dept. Matematica per le Decisioni, University of Florence (2004)
15.
Zurück zum Zitat Goll, T., Rüschendorf, L.: Minimax and minimal distance martingale measures and their relationship to portfolio optimization. Finance Stochast. 5, 557–581 (2001)CrossRef Goll, T., Rüschendorf, L.: Minimax and minimal distance martingale measures and their relationship to portfolio optimization. Finance Stochast. 5, 557–581 (2001)CrossRef
16.
Zurück zum Zitat Hernández-Hernández, D., Schied, A.: Robust utility maximization in a stochastic factor model. Stat. Decis. 24, 109–125 (2006)CrossRef Hernández-Hernández, D., Schied, A.: Robust utility maximization in a stochastic factor model. Stat. Decis. 24, 109–125 (2006)CrossRef
17.
Zurück zum Zitat Hernández-Hernández, D., Schied, A.: A control approach to robust utility maximization with logarithmic utility and time-consistent penalties. Stoch. Process. Appl. 117, 980–1000 (2007)CrossRef Hernández-Hernández, D., Schied, A.: A control approach to robust utility maximization with logarithmic utility and time-consistent penalties. Stoch. Process. Appl. 117, 980–1000 (2007)CrossRef
18.
Zurück zum Zitat Hodges, S.D., Neuberger, A.: Optimal replication of contingent claims under transaction costs. Rev. Futures Markets 8, 222–239 (1989) Hodges, S.D., Neuberger, A.: Optimal replication of contingent claims under transaction costs. Rev. Futures Markets 8, 222–239 (1989)
19.
Zurück zum Zitat Jacod, J.: Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics, vol. 714. Springer, Berlin (1979) Jacod, J.: Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics, vol. 714. Springer, Berlin (1979)
20.
Zurück zum Zitat Jacod, J.: Intégrales stochastiques par rapport à une semi-martingale vectorielle et changements de filtration. In: Séminaire de Probabilités XIV. Lecture Notes in Mathematics, vol. 784, pp. 161–172. Springer, Berlin (1980) Jacod, J.: Intégrales stochastiques par rapport à une semi-martingale vectorielle et changements de filtration. In: Séminaire de Probabilités XIV. Lecture Notes in Mathematics, vol. 784, pp. 161–172. Springer, Berlin (1980)
21.
Zurück zum Zitat Kabanov, Y.M., Stricker, C.: On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper. Math. Finance 12, 125–134 (2002)CrossRef Kabanov, Y.M., Stricker, C.: On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper. Math. Finance 12, 125–134 (2002)CrossRef
22.
Zurück zum Zitat Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999)CrossRef Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999)CrossRef
23.
Zurück zum Zitat Kramkov, D., Schachermayer, W.: Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13, 1504–1516 (2003)CrossRef Kramkov, D., Schachermayer, W.: Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13, 1504–1516 (2003)CrossRef
24.
Zurück zum Zitat Mania, M., Schweizer, M.: Dynamic exponential utility indifference valuation. Ann. Appl. Probab. 15, 2113–2143 (2005)CrossRef Mania, M., Schweizer, M.: Dynamic exponential utility indifference valuation. Ann. Appl. Probab. 15, 2113–2143 (2005)CrossRef
25.
Zurück zum Zitat Müller, M.: Market Completion and Robust Utility Maximization. Ph.D. thesis, Humboldt Universität zu Berlin (2005) Müller, M.: Market Completion and Robust Utility Maximization. Ph.D. thesis, Humboldt Universität zu Berlin (2005)
26.
Zurück zum Zitat Owari, K.: Robust exponential hedging in a Brownian setting. JSIAM Lett. 1, 64–67 (2009) Owari, K.: Robust exponential hedging in a Brownian setting. JSIAM Lett. 1, 64–67 (2009)
27.
Zurück zum Zitat Owari, K.: A note on utility maximization with unbounded random endowment. Asia-Pacific Financial Markets. doi: 10.1007/s10690-010- 9122-4 (2010, forthcoming) Owari, K.: A note on utility maximization with unbounded random endowment. Asia-Pacific Financial Markets. doi: 10.1007/s10690-010- 9122-4 (2010, forthcoming)
28.
Zurück zum Zitat Owari, K.: Robust exponential hedging and indifference valuation. Int. J. Theor. Appl. Finance (2010, forthcoming) Owari, K.: Robust exponential hedging and indifference valuation. Int. J. Theor. Appl. Finance (2010, forthcoming)
29.
Zurück zum Zitat Owen, M.P., Žitković, G.: Optimal investment with an unbounded random endowment and utility-based pricing. Math. Finance 19, 129–159 (2009)CrossRef Owen, M.P., Žitković, G.: Optimal investment with an unbounded random endowment and utility-based pricing. Math. Finance 19, 129–159 (2009)CrossRef
30.
Zurück zum Zitat Quenez, M.-C.: Optimal portfolio in a multiple-priors model. In: Dalang, R.C., Dozzi, M., Russo, F. (eds.) Seminar on Stochastic Analysis, Random Fields and Applications IV. Progr. Probab., vol. 58, pp. 291–321. Birkhäuser, Basel (2004) Quenez, M.-C.: Optimal portfolio in a multiple-priors model. In: Dalang, R.C., Dozzi, M., Russo, F. (eds.) Seminar on Stochastic Analysis, Random Fields and Applications IV. Progr. Probab., vol. 58, pp. 291–321. Birkhäuser, Basel (2004)
31.
Zurück zum Zitat Rouge, R., El Karoui, N.: Pricing via utility maximization and entropy. Math. Finance 10, 259–276 (2000) Rouge, R., El Karoui, N.: Pricing via utility maximization and entropy. Math. Finance 10, 259–276 (2000)
32.
Zurück zum Zitat Schachermayer, W.: Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Probab. 11, 694–734 (2001)CrossRef Schachermayer, W.: Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Probab. 11, 694–734 (2001)CrossRef
33.
Zurück zum Zitat Schachermayer, W.: A super-martingale property of the optimal portfolio process. Finance Stochast. 7, 433–456 (2003)CrossRef Schachermayer, W.: A super-martingale property of the optimal portfolio process. Finance Stochast. 7, 433–456 (2003)CrossRef
34.
Zurück zum Zitat Schied, A.: Optimal investment for risk- and ambiguity-averse preferences: a duality approach. Finance Stochast. 11, 107–129 (2007)CrossRef Schied, A.: Optimal investment for risk- and ambiguity-averse preferences: a duality approach. Finance Stochast. 11, 107–129 (2007)CrossRef
35.
Zurück zum Zitat Schied, A., Wu, C.-T.: Duality theory for optimal investment under model uncertainty. Stat. Decis. 23, 199–217 (2005)CrossRef Schied, A., Wu, C.-T.: Duality theory for optimal investment under model uncertainty. Stat. Decis. 23, 199–217 (2005)CrossRef
36.
Zurück zum Zitat Simons, S.: From Hahn–Banach to monotonicity, 2nd edn. Lecture Notes in Mathematics, vol. 1693. Springer, Berlin (2008) Simons, S.: From Hahn–Banach to monotonicity, 2nd edn. Lecture Notes in Mathematics, vol. 1693. Springer, Berlin (2008)
37.
Zurück zum Zitat Wittmüss, W.: Robust optimization of cousumption with random endowment. Stochastics 80, 459–475 (2008) Wittmüss, W.: Robust optimization of cousumption with random endowment. Stochastics 80, 459–475 (2008)
38.
Zurück zum Zitat Yor, M.: Sous-espaces denses dans L 1 ou H 1 et representation des martingales. In: Séminaire de Probabilités XII. Lecture Notes in Mathematics, vol. 649, pp. 265–309. Springer, Berlin (1978) Yor, M.: Sous-espaces denses dans L 1 ou H 1 et representation des martingales. In: Séminaire de Probabilités XII. Lecture Notes in Mathematics, vol. 649, pp. 265–309. Springer, Berlin (1978)
Metadaten
Titel
Robust utility maximization with unbounded random endowment
verfasst von
Keita Owari
Copyright-Jahr
2011
Verlag
Springer Japan
DOI
https://doi.org/10.1007/978-4-431-53883-7_7

Premium Partner