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Sequential $\delta$-optimal consumption and investment for stochastic volatility markets with unknown parameters

Author

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  • Belkacem Berdjane

    (LMRS - Laboratoire de Mathématiques Raphaël Salem - UNIROUEN - Université de Rouen Normandie - NU - Normandie Université - CNRS - Centre National de la Recherche Scientifique)

  • Sergei Pergamenshchikov

    (LMRS - Laboratoire de Mathématiques Raphaël Salem - UNIROUEN - Université de Rouen Normandie - NU - Normandie Université - CNRS - Centre National de la Recherche Scientifique, Department of Mathematics and Mechanics - Tomsk State University [Tomsk])

Abstract

We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic volatility and unknown stock appreciation rate. The volatility parameter is driven by an external economic factor modeled as a diffusion process of Ornstein-Uhlenbeck type with unknown drift. We use the dynamical programming approach and find an optimal financial strategy which depends on the drift parameter. To estimate the drift coefficient we observe the economic factor $Y$ in an interval $[0,T_0]$ for fixed $T_0>0$, and use sequential estimation. We show, that the consumption and investment strategy calculated through this sequential procedure is $\delta$-optimal.

Suggested Citation

  • Belkacem Berdjane & Sergei Pergamenshchikov, 2012. "Sequential $\delta$-optimal consumption and investment for stochastic volatility markets with unknown parameters," Working Papers hal-00743164, HAL.
    • Handle: RePEc:hal:wpaper:hal-00743164
    Note: View the original document on HAL open archive server: https://hal.science/hal-00743164v2
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    References listed on IDEAS

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    1. repec:hum:wpaper:sfb649dp2006-007 is not listed on IDEAS
    2. Thaleia Zariphopoulou, 2001. "A solution approach to valuation with unhedgeable risks," Finance and Stochastics, Springer, vol. 5(1), pages 61-82.
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