Utility maximization in a multidimensional semimartingale model with nonlinear wealth dynamics

• Published in: Mathematics and financial economics Vol. 15; no. 4; pp. 775 - 809
• Main Authors: Junca, Mauricio, Serrano, Rafael
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2021; Springer Nature B.V
• Subjects: Allocations; Applications of Mathematics; Business models; Cash flow; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Finance; Insurance; Macroeconomics/Monetary Economics//Financial Economics; Management; Martingales; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinear dynamics; Optimization; Quantitative Finance; Risk aversion; Statistics for Business; Technology; Verification; Wealth; Semimartingale characteristics; C02; G11; Nonlinear wealth; Convex duality; Portfolio choice; Utility maximization; Martingale method; C61
• ISSN: 1862-9679; 1862-9660
• DOI: 10.1007/s11579-021-00296-z

We explore martingale and convex duality techniques to maximize expected risk-averse utility from consumption in a general multi-dimensional (non-Markovian) semimartingale market model with jumps and non-linear wealth dynamics. The model allows to incorporate additional cash flows via non-linear margin payment functions in the drift term that depend on the allocation proportion. These can be used to cast frictions such as the impact of the portfolio choices of a ‘large’ investor on the expected assets’ returns, funding costs arising from differential borrowing and lending rates, and the cash inflow of a firm in a neoclassical economy with constant return-to-scale Cobb–Douglas technology subject to exogenous aggregate shocks. We provide a general verification theorem for random utility fields satisfying the usual Inada conditions, find conditions under which jumps in our model lead to precautionary saving, and present an explicit characterization for CRRA. We report two-fund separation-type results which assert that optimal allocations move along one-dimensional segments, and illustrate our results numerically for various margin payment functions and bounded variation tempered α -stable jumps.