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Optimal investment and consumption with transaction costs. (English) Zbl 0813.60051
The underlying continuous-time Merton model with constant market coefficients was studied in the presence of linear transaction costs by M. H. A. Davis and A. R. Norman [Math. Oper. Res. 15, No. 4, 676-713 (1990; Zbl 0717.90007)]. The utility function is a concave power function. The present long paper removes some restrictive and hardly verifiable assumptions of Davis and Norman replacing them by the sole assumption of a finite value function. The nature of the optimal investment and consumption policy is now fully understood describing the policy by singularly continuous processess. The stock price is described by a geometric Brownian motion where the mean rate of return for the stock price is larger than the interest rate. Thus the HJB equation is of second order. By the principle of dynamic programming the value function is shown to be a viscosity solution. The concavity of the value function provides useful regularity, and even the continuity of the second partial derivatives can be shown. Then bounds are obtained for the endpoints of the interval in which the optimal proportion of wealth lies. The paper is written for the reader who is not familiar with viscosity solutions. The presentation of this concept is self-contained and tailor-made for the present problem. In the appendix the sensitivity of the value function is studied relative to the transaction cost.
Reviewer: M.Schäl (Bonn)

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
91G10 Portfolio theory
93E20 Optimal stochastic control
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