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Optimal Investment Problems with Marked Point Processes

  • Claudia CeciEmail author
Conference paper
Part of the Progress in Probability book series (PRPR, volume 63)

Abstract

Optimal investment problems in an incomplete financial market with pure jump stock dynamics are studied. An investor with Constant Relative Risk Aversion (CRRA) preferences, including the logarithmic utility, wants to maximize her/his expected utility of terminal wealth by investing in a bond and in a risky asset. The risky asset price is modeled as a geometric marked point process, whose dynamics is driven by two independent Poisson processes, describing upwards and downwards jumps. A stochastic control approach allows us to provide optimal investment strategies and closed formulas for the value functions associated to the utility optimization problems. Moreover, the solution to the dual problems associated to the utility maximization problems are derived. The case when intermediate consumption is allowed is also discussed.

Keywords

Utility Maximization Pure Jump processes Optimal Stochastic Control 

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References

  1. 1.
    N. Bellamy, Wealth optimization in an incomplete market driven by a jump-diffusion process, Journal of Mathematical Economics, 35 (2001), 259287.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    F. Bellini and M. Frittelli, On the existence of minimax martingale measures, Mathematical Finance, 12 (2002), 121.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    S. Biagini and M. Frittelli, Utility maximization in incomplete markets for unbounded processes, Finance and Stochastics, 9 (2005), 493517.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    F.E. Benth, K.H. Karlsen, and K. Reikvam, Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: a viscosity solution approach, Finance and Stochastics, 5 (3) (2001), 275303.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    R. Boel and P. Varaiya, Optimal control of jump processes, SIAM Journal on Control and Optimization, 15 (1) (1977), 92119.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    C. Ceci, An HJB approach to exponential utility maximization for jump processes, International Journal of Risk Assessment and Management, 11 (1/2) (2009), 104121.MathSciNetCrossRefGoogle Scholar
  7. 7.
    C. Ceci and B. Bassan, Mixed optimal stopping and stochastic control problems with semicontinuous final reward for diffusion processes, Stochastics and Stochastics Reports, 76 (4) (2004), 323337.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    M.G. Crandall and P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. A.M.S., 277 (1983), 142.Google Scholar
  9. 9.
    G. Callegaro and T. Vargiolu, Optimal portfolio for HARA utility functions in a pure jump multidimensional incomplete market, International Journal of Risk Assessment and Management, 11 (1/2) (2009), 180200.CrossRefGoogle Scholar
  10. 10.
    C. Doléans-Dade, Quelques applications de la formule de changement de variables pour le semi-martingales, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 16 (1970), 181194.zbMATHCrossRefGoogle Scholar
  11. 11.
    S.N. Ethier and T.G. Kurtz, Markov Processes: Characterization and Convergence, J. Wiley New York, 1986.Google Scholar
  12. 12.
    W. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions, New York, Springer, 1993.zbMATHGoogle Scholar
  13. 13.
    D.G. Luenberger, Optimization by Vector Space Methods, Wiley New York, 1969.Google Scholar
  14. 14.
    R. Merton, Optimal consumption and portfolio rules in a continuous time model, Journal of Economic Theory, 3 (1971), 373413.MathSciNetCrossRefGoogle Scholar
  15. 15.
    H. Pham, Optimal stopping of controlled jump diffusion processes: a viscosity solution approach, Journal of Mathematical System, Estimation, and Control, 8 (1) (1998),127.MathSciNetGoogle Scholar
  16. 16.
    H. Pham, Smooth solution to optimal investment models with stochastic volatilities and portfolio constraints, Appl. Math. Optim., 78 (2002), 5578.MathSciNetCrossRefGoogle Scholar
  17. 17.
    W. Schachermayer, Utility maximization in incomplete market, in: Stochastic Methods in Finance, M. Frittelli and W.J. Runggaldier, Eds., Springer-Verlag, 2004, 255293.Google Scholar
  18. 18.
    T. Zariphopoulou, Consumption investment models with constraints, SIAM Journal on Control and Optimization, 30 (1994), 5984.MathSciNetCrossRefGoogle Scholar
  19. 19.
    T. Zariphopoulou, A solution approach to valuation with unhedgeable risks, Finance and Stochastics, 5 (2001), 6182.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Dipartimento di Scienze Facoltà di EconomiaUniversità di Chieti-PescaraPescaraItaly

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