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UTILITY MAXIMIZATION IN A PURE JUMP MODEL WITH PARTIAL OBSERVATION

Published online by Cambridge University Press:  02 November 2010

Paola Tardelli
Paola Tardelli
Affiliation:
Department of Electrical and Information Engineering, University of L'Aquila, 67100 L'Aquila, Italy E-mail: [email protected]
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Abstract

This article considers the asset price movements in a financial market when risky asset prices are modeled by marked point processes. Their dynamics depend on an underlying event arrivals process—a marked point process having common jump times with the risky asset price process. The problem of utility maximization of terminal wealth is dealt with when the underlying event arrivals process is assumed to be unobserved by the market agents using, as the main tool, backward stochastic differential equations. The dual problem is studied. Explicit solutions in a particular case are given.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 25 , Issue 1 , January 2011 , pp. 29 - 54
Copyright
Copyright © Cambridge University Press 2011

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References

1.Becherer, D. (2006). Bounded solutions to backward SDE's with jumps for utility optimization and indifference hedging. Annals of Applied Probability 16(4): 2027–2054.Google Scholar
2.Bellini, F. & Frittelli, M. (2002). On the existence of minimax martingale measures. Mathematical Finance 12(1): 1–21.Google Scholar
3.Brémaud, P. (1981). Point processes and queues. Martingale dynamics. New York: Springer-Verlag.CrossRefGoogle Scholar
4.Carbone, R., Ferrario, B., & Santacroce, M. (2007). Backward stochastic differential equations driven by càdlàg martingales, Theory of Probability and its Applications 52: 304–314.Google Scholar
5.Ceci, C. (2006). Risk minimizing hedging for a partially observed high frequency data model. Stochastics 78(1): 13–31.Google Scholar
6.Ceci, C. & Gerardi, A. (2009). Pricing for geometric marked point processes under partial information: Entropy approach. International Journal of Theoretical and Applied Finance 12(2): 179–207.Google Scholar
7.Ceci, C. & Gerardi, A. (2009). Indifference valuation via backward SDE's driven by Poisson martingales. Technical report R-2009-001, Dipartimento di Scienze, Universita di Chieti-Pescara.Google Scholar
8.Centanni, S. & Minozzo, M. (2006). A monte carlo approach to filtering for a class of marked doubly stochastic poisson processes. Journal of the American Statistical Association 101(476): 1582–1597.Google Scholar
9.Centanni, S. & Minozzo, M. (2006). Estimation and filtering by reversible jump MCMC for a doubly stochastic poisson model for ultra-high-frequency financial data. Statistical Modeling 6(2): 97–118.CrossRefGoogle Scholar
10.Delbaen, F., Grandits, P., Rheinlander, T., Samperi, D., Schweizer, M., & Stricker, C. (2002). Exponential hedging and entropic penalties. Mathematical Finance 12(2): 99–123.Google Scholar
11.Dellacherie, C. & Meyer, P.A. (1978). Probabilities and potential. Amsterdam: North-Holland.Google Scholar
12.Ethier, S.N. & Kurtz, T.G. (1986). Markov processes: Characterization and convergence. New York: Wiley.CrossRefGoogle Scholar
13.Frey, R. (2000). Risk minimization with incomplete information in a model for high-frequency data. Mathematical Finance 10(2): 215–225.Google Scholar
14.Frittelli, M. (2000). The minimal entropy martingale measure and the valuation problem in incomplete markets. Mathematical Finance 10(1): 39–52.Google Scholar
15.Frittelli, M. (2000). Introduction to a theory of value coherent with no-arbitrage principle. Finance and Stochastics 4(3): 275–297.Google Scholar
16.Gerardi, A. & Tardelli, P. (2006). Filtering on a partially observed ultra-high-frequency data model. Acta Applicanda Mathematicae 91(2): 193–205.CrossRefGoogle Scholar
17.Gerardi, A. & Tardelli, P. (2009). Stochastic control methods: Hedging in a market described by pure jump processes. Acta Applicanda Mathematicae, doi: 10.1007/s10440-009-9543-0Google Scholar
18.Gerardi, A. & Tardelli, P. (2010). Risk-neutral measures and pricing for a pure jump price process: A stochastic control approach. Probability in the Engineering and Informational Sciences 24(1): 47–76.Google Scholar
19.Grandits, P. & Rheinlander, T. (2002). On the minimal entropy martingale measure. Annals of Probability 30(3): 1003–1038.Google Scholar
20.Hu, Y., Imkeller, P., & Muller, M. (2005). Utility maximization in incomplete markets. Annals of Applied Probability 15(3): 1691–1712.CrossRefGoogle Scholar
21.Jacod, J. (1979). Calcul stochastique et problèmes de martingale. Berlin: Springer-Verlag.Google Scholar
22.Kirch, M. & Runggaldier, W. J. (2004). Efficient hedging when asset prices follow a geometric poisson process with unknown intensities. SIAM Journal on Control and Optimization 43(4): 1174–1195.Google Scholar
23.Kramkov, D. & Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Annals of Applied Probability 9: 904–950.CrossRefGoogle Scholar
24.Lim, T. & Quenez, N.C. (2009). Utility maximization in incomplete market with default. Preprints Equipe MATHFI PRONUM. Available from http://mathfi.math.univ-paris-diderot.fr/en/publications, or Eprint arXiv: 0811.475Google Scholar
25.Mania, M. & Santacroce, M. (2010). Exponential utility maximization under partial information. Finance and Stochastics 14: 419–448.Google Scholar
26.Morlais, M.A. (2009). Utility maximization in a jump market model. Stochastics 81(1): 1–27.Google Scholar
27.Rouge, R. & El Karoui, N. (2000). Pricing via utility maximization and entropy. Mathematical Finance 10(2): 259–276.Google Scholar
28.Schachermayer, W. (2001). Optimal investment in incomplete markets when wealth may become negative. Annals of Applied Probability 11(3): 694–734.CrossRefGoogle Scholar
29.Schweizer, M. (2001). Handbooks in Mathematical Finance. Cambridge: Cambridge University Press, 538–574.Google Scholar
30.Zariphopoulou, T. (2001). A solution approach to valuation with unhedgeable risks. Finance and Stochastics 5(1): 61–82.Google Scholar