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A remark on smooth solutions to a stochastic control problem with a power terminal cost function and stochastic volatilities

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  • Yalc{c}in Aktar
  • Erik Taflin

Abstract

Incomplete financial markets are considered, defined by a multi-dimensional non-homogeneous diffusion process, being the direct sum of an It\^{o} process (the price process), and another non-homogeneous diffusion process (the exogenous process, representing exogenous stochastic sources). The drift and the diffusion matrix of the price process are functions of the time, the price process itself and the exogenous process. In the context of such markets and for power utility functions, it is proved that the stochastic control problem consisting of optimizing the expected utility of the terminal wealth, has a classical solution (i.e. $C^{1,2}$). This result paves the way to a study of the optimal portfolio problem in incomplete forward variance stochastic volatility models, along the lines of Ref: Ekeland et al.

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  • Yalc{c}in Aktar & Erik Taflin, 2014. "A remark on smooth solutions to a stochastic control problem with a power terminal cost function and stochastic volatilities," Papers 1405.3566, arXiv.org.
    • Handle: RePEc:arx:papers:1405.3566
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    References listed on IDEAS

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    1. Ivar Ekeland & Erik Taflin, 2003. "A theory of bond portfolios," Papers math/0301278, arXiv.org, revised May 2005.
    2. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    3. repec:dau:papers:123456789/6041 is not listed on IDEAS
    4. Hans Buehler, 2006. "Consistent Variance Curve Models," Finance and Stochastics, Springer, vol. 10(2), pages 178-203, April.
    5. Tomas Björk & Lars Svensson, 2001. "On the Existence of Finite‐Dimensional Realizations for Nonlinear Forward Rate Models," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 205-243, April.
    6. {L}ukasz Delong & Claudia Kluppelberg, 2008. "Optimal investment and consumption in a Black--Scholes market with L\'evy-driven stochastic coefficients," Papers 0806.2570, arXiv.org.
    7. Hans Buehler, 2006. "Consistent Variance Curve Models," Finance and Stochastics, Springer, vol. 10(2), pages 178-203, April.
    8. Thaleia Zariphopoulou, 2001. "A solution approach to valuation with unhedgeable risks," Finance and Stochastics, Springer, vol. 5(1), pages 61-82.
    9. Belkacem Berdjane & Serguei Pergamenshchikov, 2013. "Optimal consumption and investment for markets with random coefficients," Finance and Stochastics, Springer, vol. 17(2), pages 419-446, April.
    10. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
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    Cited by:

    1. John R. Birge & Lijun Bo & Agostino Capponi, 2018. "Risk-Sensitive Asset Management and Cascading Defaults," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 1-28, February.
    2. Dariusz Zawisza, 2016. "Smooth solutions to discounted reward control problems with unbounded discount rate and financial applications," Papers 1602.00899, arXiv.org, revised Feb 2016.

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