The Annals of Applied Probability

Bounded solutions to backward SDEs with jumps for utility optimization and indifference hedging

Dirk Becherer

Full-text: Open access

Abstract

We prove results on bounded solutions to backward stochastic equations driven by random measures. Those bounded BSDE solutions are then applied to solve different stochastic optimization problems with exponential utility in models where the underlying filtration is noncontinuous. This includes results on portfolio optimization under an additional liability and on dynamic utility indifference valuation and partial hedging in incomplete financial markets which are exposed to risk from unpredictable events. In particular, we characterize the limiting behavior of the utility indifference hedging strategy and of the indifference value process for vanishing risk aversion.

Article information

Source
Ann. Appl. Probab., Volume 16, Number 4 (2006), 2027-2054.

Dates
First available in Project Euclid: 17 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1169065215

Digital Object Identifier
doi:10.1214/105051606000000475

Mathematical Reviews number (MathSciNet)
MR2288712

Zentralblatt MATH identifier
1132.91457

Subjects
Primary: 60G57: Random measures 60H30: Applications of stochastic analysis (to PDE, etc.) 91B28
Secondary: 60H99: None of the above, but in this section 60G44: Martingales with continuous parameter 60G55: Point processes

Keywords
Backward stochastic differential equations random measures utility optimization dynamic indifference valuation incomplete markets hedging entropy

Citation

Becherer, Dirk. Bounded solutions to backward SDEs with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16 (2006), no. 4, 2027--2054. doi:10.1214/105051606000000475. https://projecteuclid.org/euclid.aoap/1169065215


Export citation

References

  • Barles, G., Buckdahn, R. and Pardoux, E. (1997). Bsde's and integral-partial differential equations. Stochastics Stochastic Rep. 60 57–83.
  • Becherer, D. (2003). Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insurance Math. Econ. 33 1–28.
  • Becherer, D. (2004). Utility-indifference hedging and valuation via reaction diffusion systems. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 27–51.
  • Becherer, D. and Schweizer, M. (2005). Classical solutions to reaction diffusion systems for hedging problems with interacting Itô and point processes. Ann. Appl. Probab. 15 1111–1144.
  • Benth, F. and Meyer-Brandis, T. (2005). The density process of the minimal entropy martingale measure in a stochastic volatility model with jumps. Finance and Stochastics 9 563–575.
  • Bielecki, T. and Jeanblanc, M. (2007). Indifference pricing of defaultable claims. In Indifference Pricing (R. Carmona, ed.). Princeton Univ. Press. To appear. Available at http://www.maths.univ-evry.fr/pages_perso/jeanblanc/pubs/bj-indif.pdf.
  • Bielecki, T., Jeanblanc, M. and Rutkowski, M. (2004). Hedging of defaultable claims. Paris-Princeton Lectures on Mathematical Finance 2003. Lecture Notes in Math. 1847 1–132. Springer, Berlin.
  • Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M. and Stricker, C. (2002). Exponential hedging and entropic penalties. Math. Finance 12 99–123.
  • El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 1 1–71.
  • Foldes, L. (2000). Valuation and martingale properties of shadow prices: An exposition. J. Econom. Dynam. Control 24 1641–1701.
  • Frittelli, M. (2000). Introduction to a theory of value coherent with the no-arbitrage principle. Finance and Stochastics 4 275–297.
  • Grandits, P. and Rheinländer, T. (2002). On the minimal entropy martingale measure. Ann. Probab. 30 1003–1038.
  • He, S., Wang, J. and Yan, J. (1992). Semimartingale Theory and Stochastic Calculus. Science Press, Beijing.
  • Hu, Y., Imkeller, P. and Müller, M. (2005). Utility maximization in incomplete markets. Ann. Appl. Probab. 15 1691–1712.
  • Jacod, J. and Shiryaev, A. (2003). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • Kabanov, Y. and Stricker, C. (2002). The optimal portfolio for the exponential utility maximization: Remarks to the six-authors paper. Math. Finance 12 125–134.
  • Kazamaki, N. (1994). Continuous Exponential Martingales and BMO. Springer, Berlin.
  • Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558–602.
  • Kramkov, D. and Sirbu, M. (2006). Asymptotic analysis of utility-based hedging strategies for small numbers of contingent claims. Preprint, Carnegie Mellon Univ. Available at http://www.math.cmu.edu/~kramkov/publications/hedging_06.pdf.
  • Lepingle, D. and Memin, J. (1978). Sur l'intégrabilité uniforme des martingales exponentielles. Z. Wahrsch. Verw. Gebiete 42 175–203.
  • Mania, M. and Schweizer, M. (2005). Dynamic exponential utility indifference valuation. Ann. Appl. Probab. 15 2113–2143.
  • Rouge, R. and El Karoui, N. (2000). Pricing via utility maximization and entropy. Math. Finance 10 259–276.
  • Schweizer, M. (2001). A guided tour through quadratic hedging approaches. In Option Pricing, Interest Rates and Risk Management (E. Jouini, J. Cvitanić and M. Musiela, eds.) 538–574. Cambridge Univ. Press.
  • Tang, S. and Li, X. (1994). Necessary conditions for optimal control for stochastic systems with random jumps. SIAM J. Control Optim. 32 1447–1475.