Asymptotic analysis of optimal investment and consumption with transaction costs. (Q1887271)

The authors consider an agent seeking the optimally invest and consume in the presence of proportional transaction costs. The agent can invest in a stock modelled as a geometric Brownian motion and in a money market with constant rate of interest. He may also consume and get utility \(U(c)=c^{1-p}/(1-p)\), where \(p>0,\;p\neq 1\). In addition, the agent must pay a proportional transaction costs \(\lambda >0\) for transferring capital between the stock and the money market. All consumptions are done from the money market. The agent wishes to maximize the expected discounted integral over \([0,\infty)\) of the utility of consumption. To solve the problem, the authors heuristically derive several terms of a power series expansion of a value function in powers of \(\lambda^{1/3}\). Then the rigorous expansion is presented. The asymptotic results on the boundary of the ``no-trade'' region are obtained. The key results are proved using viscosity sub- and supersolutions.