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An asymptotic expansion scheme for optimal investment problems. (English) Zbl 1181.91302

Summary: We shall propose a new computational scheme for the evaluation of the optimal portfolio for investment. Our method is based on an extension of the asymptotic expansion approach which has been recently developed for pricing problems of the contingent claims’ analysis by N. Kunitomo and A. Takahashi [“Pricing average options”, Jpn. Financ. Rev. 14, 1–20 (1992); Math. Finance 11, No. 1, 117–151 (2001; Zbl 0994.91023); Ann. Appl. Probab. 13, No. 3, 914–952 (2003; Zbl 1091.91037)], N. Yoshida [J. Jap. Stat. Soc. 22, No. 2, 139–159 (1992; Zbl 0778.62018)], A. Takahashi [“Essays on the valuation problems of contingent claims”, unpublished Ph.D. dissertation, Haas School of Business, University of California, Berkeley (1995); Asia-Pac. Financ. Mark. 6, No. 2, 115–151 (1999; Zbl 1153.91568)] and A. Takahashi and N. Yoshida [“Monte Carlo simulation with asymptotic method”, preprint (2001)]. In particular, we will explicitly derive a formula of the optimal portfolio associated with maximizing utility from terminal wealth in a financial market with Markovian coefficients, and give a numerical example for a power utility function.

MSC:

91G10 Portfolio theory
91G60 Numerical methods (including Monte Carlo methods)
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