Homogenization and asymptotics for small transaction costs (Q2862451)

The authors study the classical Merton problem of lifetime consumption-portfolio optimization with small proportional transaction costs. A natural approach to the solution of Merton's problem is to obtain an asymptotic expansion of the value function in terms of the small transaction costs. In this connection, a unified approach to the asymptotic analysis of the problem is developed. More precisely, let the value function of the Merton infinite-horizon optimal consumption-portfolio problem with zero transaction costs be denoted by \(v(s,z)\), where \(z\) is the initial capital and \(s\) is the initial value of the risky asset given by a time-homogeneous stochastic differential equation. The value function for the problem with transaction costs is a function of \(s\) and the pair \((x,y)\) representing the wealth in the saving and in the stock accounts, respectively. The total wealth is simply given by \(x+y\). For a small proportional transaction cost \(\varepsilon^3>0\) let \(v^{\varepsilon}(s,x,y)\) be the maximum expected discounted utility from consumption. The main analytical objective is to obtain an expansion for \(v^{\varepsilon}\) in terms of \(\varepsilon>0\). The first-order term in the expansion is explicitly calculated through a singular ergodic control problem which can be solved in closed form in the one-dimensional case. The main ideas how to consider a general utility function and general dynamics for the underlying assets came from homogenization theory and from the theory of viscosity solutions.