Economy – Quantitative Finance – Computational Finance
Scientific paper
2010-10-21
Annals of Applied Probability 2010, Vol. 20, No. 4, 1341-1358
Economy
Quantitative Finance
Computational Finance
Published in at http://dx.doi.org/10.1214/09-AAP648 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst
Scientific paper
10.1214/09-AAP648
In frictionless markets, utility maximization problems are typically solved either by stochastic control or by martingale methods. Beginning with the seminal paper of Davis and Norman [Math. Oper. Res. 15 (1990) 676--713], stochastic control theory has also been used to solve various problems of this type in the presence of proportional transaction costs. Martingale methods, on the other hand, have so far only been used to derive general structural results. These apply the duality theory for frictionless markets typically to a fictitious shadow price process lying within the bid-ask bounds of the real price process. In this paper, we show that this dual approach can actually be used for both deriving a candidate solution and verification in Merton's problem with logarithmic utility and proportional transaction costs. In particular, we determine the shadow price process.
Kallsen Jan
Muhle-Karbe Johannes
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