Existence and uniqueness of solutions to a quasilinear parabolic equation with quadratic gradients in financial markets

Abstract

A quasilinear parabolic equation with quadratic gradient terms is analyzed. The equation models an optimal portfolio in so-called incomplete financial markets consisting of risky assets and non-tradable state variables. Its solution allows to compute an optimal portfolio strategy. The quadratic gradient terms are essentially connected to the assumption that the so-called relative risk aversion function is not logarithmic. The existence of weak global-in-time solutions in any dimension is shown under natural hypotheses. The proof is based on the monotonicity method of Frehse. Furthermore, the uniqueness of solutions is shown under a smallness condition on the derivatives of the covariance (“diffusion”) matrices using a nonlinear test function technique developed by Barles and Murat. Finally, the influence of the non-tradable state variables on the optimal value function is illustrated by a numerical example in three dimensions.