Article Optimal Investment and Consumption for Multidimensional Spread Financial Markets with Logarithmic Utility Sahar Albosaily 1 and Serguei Pergamenchtchikov 2,3, *   Citation: Albosaily, S.; Pergamenchtchikov, S. Optimal Investment and Consumption for Multidimensional Spread Financial Markets with Logarithmic Utility. Stats 2021, 4, 1012–1026. https:// doi.org/10.3390/stats4040058 Academic Editor: Wei Zhu Received: 22 October 2021 Accepted: 26 November 2021 Published: 29 November 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Mathematics, Faculty of Science, University of Ha’il, P.O. Box 2440 Ha’il, Saudi Arabia; s.albosaily@uoh.edu.sa 2 Laboratoire de Mathématiques Raphael Salem, UMR 6085 CNRS-Université de Rouen, F76801 Saint-Étienne-du-Rouvray, France 3 International Laboratory of Statistics of Stochastic Processes and Quantitative Finance, Tomsk State University, 634050 Tomsk, Russia * Correspondence: Serge.Pergamenshchikov@univ-rouen.fr Abstract: We consider a spread financial market defined by the multidimensional Ornstein–Uhlenbeck (OU) process. We study the optimal consumption/investment problem for logarithmic utility func- tions using a stochastic dynamical programming method. We show a special verification theorem for this case. We find the solution to the Hamilton–Jacobi–Bellman (HJB) equation in explicit form and as a consequence we construct optimal financial strategies. Moreover, we study the constructed strategies with numerical simulations. Keywords: optimality; Feynman–Kac mapping; Hamilton–Jacobi–Bellman equation; Itô formula; Brow- nian motion; Ornstein–Uhlenbeck processes; stochastic processes; financial markets; spread markets 1. Introduction In this paper, we consider an optimisation and consumption problem for spread financial markets on the time interval [0, T]. Our goal is to find investment and consumption strategies which maximise the terminal wealth and the consumption during the investment period for logarithmic utilities. The spread means the difference between two prices of co- integrated risky assets (see, for example, in [1,2]). The co-integration property for time series means that the difference between them represents a stationary process. If we consider two or more stocks that are selected for the joint integration and correlation between them, then multidimensional stationary processes should be used. In discrete time, multivariate stationary processes can be well approximated by auto-regressive models in R d X n = AX n−1 + ξ n , where (ξ n ) n≥1 are i.i.d. random zero mean vectors in R d . In continuous time, this model corresponds to the stochastic differential equation in R d (see, for example, [3,4]), i.e., dX t = AX t dt + σdW t , where W t is a standard Brownian motion. This equation is called the Ornstein–Uhlenbeck (OU) process in R d . Thus, it is very natural to use such processes to model the spread markets in continuous time with the stable matrix A, i.e., when all the eigenvalues have negative real parts. It should be noted that in this case, the spread is a mean-reverting process, therefore, if investors sell high stocks (with high prices) and buy low stocks (with low prices) when the spread widens, then this will assure the continuity of profits over time in such market models. Thus, the idea is to go long when the price is under the long-term mean and go short for the stock price over the long-term mean, as reciprocal actions for the chosen co-integrated stocks. Moreover, it should be noted that the spread Stats 2021, 4, 1012–1026. https://doi.org/10.3390/stats4040058 https://www.mdpi.com/journal/stats