Weak Convergence of Utility-Risk Portfolios

In this article, we establish the validity of the Donsker invariance principle for weak convergence of the optimal utility-downside-risk portfolio payoff. Thus, the optimal solution is robust to changes in the market model via the pricing kernel. The convergence result is valid for all feasible combinations of utility functions satisfying the Inada conditions, convex downside deviation risk measures, and pricing kernels satisfying integrability conditions. Thus, our results provide a unified framework to understand whether utility-downside-risk solutions are stable under mild misspecifications of asset price processes. As another application, we further obtain a result that the optimal utility-risk values and policies driven by discrete-time binomial models converge weakly to those of the limiting continuous-time Black-Scholes model. We demonstrate the convergence result using numerical examples and find that the optimal utility-risk value converges as the time interval of the binomial model gets smaller.