Asymptotic analysis for Merton's problem with transaction costs in power utility case

Abstract

We revisit the optimal investment and consumption problem with proportional transaction costs. We prove that both the value function and the slopes of the lines demarcating the no-trading region are analytic functions of cube root of the transaction cost parameter. Also, we can explicitly calculate the coefficients of the fractional power series expansions of the value function and the no-trading region.

Keywords::

• stochastic control
• utility maximization
• transaction costs
• shadow price
• asymptotic analysis

AMS Subject Classification::

• 91G10
• 91B16

Notes

^1. As discussed in [4], by scaling the stock price, we can easily see that this formulation is equivalent to the model with bid and ask price of and , for constants and .

^2. The case corresponds to the case of end-point singularity in [1] (Proposition 6.9(2)).