Asymptotic analysis of the expected utility maximization problem with respect to perturbations of the numéraire

Abstract

In an incomplete model, where under an appropriate numéraire, the stock price process is driven by a sigma-bounded semimartingale, we investigate the behavior of the expected utility maximization problem under small perturbations of the numéraire.We establish a quadratic approximation of the value function and a first-order expansion of the terminal wealth. Relying on a description of the base return process in terms of its semimartingale characteristics, we also construct wealth processes and nearly optimal strategies that allow for matching the primal value function up to the second order. We also link perturbations of the numéraire to distortions of the finite-variation part and martingale part of the stock price return and characterize the asymptotic expansions in terms of the risk-tolerance wealth process.Date: February 11, 2020.2010 Mathematics Subject Classification. 91G10, 93E20. JEL Classification: C61, G11. Key words and phrases. Sensitivity analysis, stability, utility maximization, optimal investment, risktolerance process, arbitrage of the first kind, no unbounded profit with bounded risk, local martingale deflator, duality theory, semimartingale, incomplete market, sigma-bounded process, semimartingale characteristics, Sherman-Morrison formula, envelope theorem, numéraire.

Stability and asymptotic analysis of the Föllmer-Schweizer decomposition on a finite probability space

“…Mathematically, (1) is also closely related to the problem considered in [Sch94], finding the best approximation of a random variable by a stochastic integral (plus a constant). Quadratic optimization problems of the form (1) also appear in the asymptotic analysis of stochastic control problems with respect to perturbations of the initial data, where they govern the second-order correction terms, see [KS06a], [KS06b], [MS19], and [Mos20] for details.…”