Continuity of Utility Maximization under Weak Convergence

Abstract

In this paper we find tight sufficient conditions for the continuity of the value of the utility maximization problem from terminal wealth with respect to the convergence in distribution of the underlying processes. We also establish a weak convergence result for the terminal wealths of the optimal portfolios. Finally, we apply our results to the computation of the minimal expected shortfall (shortfall risk) in the Heston model by building an appropriate lattice approximation.2010 Mathematics Subject Classification. 91G10, 91G20.

Convergence of optimal expected utility for a sequence of discrete‐time markets

“…The incomplete‐market case on which we focus has recently been treated in a setting of greater generality by Bayraktar, Dolinsky, and Guo (2018). 1 Their paper assumes that the financial markets are general semi‐martingales and the limiting market S is a continuous semi‐martingale.…”

“…Also, the utility function U in Bayraktar et al. (2018) may measurably depend on the observed trajectory of the stock price. Hence, their model includes our special and paradigmatic case, where is induced by a single (scaled) random variable ζ and S is geometric Brownian motion.…”

“…The key assumptions in Bayraktar et al. (2018) are Assumption 2.3(ii), that a certain family of random variables is uniformly integrable, and Assumption 2.5, which effectively assumes away the possibility of asymptotic arbitrage. Lemma 2.2 of Bayraktar et al.…”

“…Lemma 2.2 of Bayraktar et al. (2018) provides some fairly strong conditions under which Assumption 2.3(ii) is satisfied, conditions that are not related to the concept of asymptotic elasticity. In comparison, we deduce the uniform integrability of certain corresponding families of dual random variables (see Equations 12 and 18) from the assumption that U has asymptotic elasticity less than 1.…”

“…11This paper (Bayraktar et al., 2018) was put on arXiv in November 2018; the authors kindly brought their paper to our attention after a first version of the current paper appeared on arXiv in July 2019. The references to their results refer to the arXiv version of Bayraktar et al.…”