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

✍ Scribed by Bertram Düring; Ansgar Jüngel

• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 379 KB
• Volume: 62
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2005.03.068

✦ Synopsis

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.