An Approximation Scheme for Solution to the Optimal Investment Problem in Incomplete Markets

Abstract

We provide an approximation scheme for the maximal expected utility and optimal investment policies for the portfolio choice problem in an incomplete market. Incompleteness stems from the presence of a stochastic factor which affects the dynamics of the correlated stock price. The scheme is built on the Trotter--Kato approximation and is based on an intuitively pleasing splitting of the Hamilton--Jacobi--Bellman (HJB) equation in two subequations. The first is the HJB equation of a portfolio choice problem with a stochastic factor but in a complete market, while the other is a linear equation corresponding to the evolution of the orthogonal (nontraded) part of the stochastic factor. We establish convergence of the scheme to the unique viscosity solution of the marginal HJB equation, and, in turn, derive a computationally tractable representation of the maximal expected utility and construct an $\varepsilon$-optimal portfolio in a feedback form.