Homogenization and asymptotics for small transaction costs

We consider the classical Merton problem of lifetime consumption-portfolio optimization with small proportional transaction costs. The first order term in the asymptotic expansion is explicitly calculated through a singular ergodic control problem which can be solved in closed form in the one-dimensional case. Unlike the existing literature, we consider a general utility function and general dynamics for the underlying assets. Our arguments are based on ideas from homogenization theory and use convergence tools from the theory of viscosity solutions. The multidimensional case is studied in our companion paper [D. Possamaï, H. M. Soner, and N. Touzi, Homogenization and Asymptotics for Small Transaction Costs: The Multidimensional Case, arXiv:1212.6275v2 [math.AP], preprint, 2012] using the same approach.