Statistics & Probability Letters 78 (2008) 426–434 FBSDE approach to utility portfolio selection in a market with random parameters $ Rene´ Ferland à , Franc - ois Watier De´partement de mathe´matiques, Universite´du Que´bec a` Montre´al, Montre´al Que., Canada H3C 3P8 Received 23 March 2006; received in revised form 14 May 2007; accepted 25 July 2007 Available online 22 August 2007 Abstract A continuous-time utility portfolio selection problem is studied in a market in which the interest rate, appreciation rates and volatility coefficients are driven by Brownian motion. We construct an optimal portfolio using results from forward–backward stochastic differential equations (FBSDE) theory. As an illustration, exact computation of the optimal strategy is done for the power and exponential type utilities. r 2007 Elsevier B.V. All rights reserved. MSC: primary 91B28; secondary 91B70; 60H10 Keywords: Expected utility maximization; Optimal portfolio; Forward–backward stochastic differential equations 1. Introduction In the field of economics, an individual’s utility curve encompasses his degree of satisfaction associated with different level of wealth attained as well as his behavior towards risk in an attempt to acquire additional wealth. Utility portfolio selection addresses the issue of allocation of wealth amongst a basket of securities (a bond and several stocks) in order to maximize the expected utility from consumption and/or terminal wealth. The first treatment of this stochastic optimization problem in a continuous-time setting originated in the seminal papers of Merton (1969, 1971). In the context of a market model with deterministic coefficients (appreciation rate, volatility and interest rate) and stock price processes driven by Brownian motion, Merton used the Hamilton–Jacobi–Bellman (HJB) equation of dynamic programming to construct an explicit optimal portfolio for log and power utility. Later on, the emergence of the concept of equivalent martingale measure in contingent claim valuation theory led Pliska (1986) and Karatzas et al. (1987) to develop an alternate method for solving utility portfolio selection problems, namely the martingale method. Basically, one uses the martingale representation theorem to guarantee the existence of a portfolio process attaining a given admissible wealth process, and further one solves a static utility maximization problem using convex analysis to obtain the optimal wealth. ARTICLE IN PRESS www.elsevier.com/locate/stapro 0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2007.07.016 $ This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada. à Corresponding author. E-mail address: ferland.rene@uqam.ca (R. Ferland).