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Portfolios and risk premia for the long run. (English) Zbl 1247.91172

The author studies the optimal strategy problem under a general multidimensional diffusion processes driven by state variables. This in particular includes stochastic volatility models with stochastic drifts. They derive a closed-form expression for the optimal portfolio when the maturity tends to infinity. Their proof is based on similar arguments developed by H. Kaise and S.-J. Cheu [Ann. Probab. 34, No. 1, 284–320 (2006; Zbl 1092.60030)], and they also provide hints on the link with the Donsker-Varadhan approach of large deviations [M. D. Donsker and S. R. S. Varadhan, Commun. Pure Appl. Math. 28, 1–47 (1975; Zbl 0323.60069); ibid. 28, 279–301 (1975; Zbl 0348.60031), Commun. Pure Appl. Math. 29, 389–461 (1976; Zbl 0348.60032), Commun. Pure Appl. Math. 36, 183-212 (1983; Zbl 0512.60068)]. They derive candidates for the optimal portfolio and risk premia, which depend on the solutions of a quasi-linear PDE (essentially a long-time version of the HJB equation). They also prove existence of the solutions of such equations.

MSC:

91G10 Portfolio theory
62P05 Applications of statistics to actuarial sciences and financial mathematics
91G20 Derivative securities (option pricing, hedging, etc.)
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