 Advances in Pure Mathematics, 2011, 1, 81-83 doi:10.4236/apm.2011.13018 Published Online May 2011 (http://www.scirp.org/journal/apm) Copyright © 2011 SciRes. APM Portfolio Optimization without the Self-Financing Assumption Moawia Alghalith Department of Economics, University of the West Indies, St. Augustine, Trinidad and Tobago E-mail: malghalith@gmail.com Received January 13, 2011; revised February 28, 2011; accepted March 5, 2011 Abstract In this paper, we relax the assumption of a self-financing strategy in the dynamic investment models. In so doing we provide smooth solutions and constrained viscosity solutions. Keywords: Portfolio, Investment, Stochastic, Viscosity Solutions, Self Financing 1. Introduction The literature on dynamic portfolio optimization is vast. However, previous literature on dynamic investment relied on the assumption of a self-financing strategy; that is, the investor cannot add or withdraw funds during the trading horizon. Examples include , ,  and  among many others. However, this assumption is somewhat restrictive and sometimes unrealistic. Moreover, even with the assumption of a self-financ- ing strategy, the previous literature usually provided explicit solutions under the assumption of a logarithmic or power utility function. Therefore, the assumption of a self- financing strategy did not offer a significant simplification of the solutions. Therefore, the self-financing assumption needs to be relaxed. Consequently, the goal of this paper is to relax the assumption of self-financing strategies. In this paper, we show that the assumption of a self-financing strategy can be relaxed without a significant complication of the optimal solutions. In so doing, we present a stochastic-fac- tor incomplete-markets investment model and provide both smooth solutions and constrained viscosity solutions. 2. The Model We consider an investment model, which includes a risky asset, a risk-free asset and a random external economic factor (see, for example, ). We use a three-dimensional standard Brownian motion1s 2s,,WW 3s ,stsTWF on the probability spaceΩ,,sFP, where 0ssTF is the augmentation of filtration. The risk-free asset price process is d0,TstrY sSe where srY 2bCRä is the rate of return and sY is the stochastic economic factor. The dynamics of the risky asset price are given by 11ddd,ssss sSSμYsσYW (1) where sμY and 1sσY are the rate of return and the volatility, respectively. The economic factor process is given by 22ddd,,ss sstYbYsσYWYy (2) where 2sσY is its volatility and 1sbYC Rä . The amount of money added to or withdrawn from the investment at time s is denoted by Φ,s and its dynamics are given by 33dΦdd,ssssaYs σYW (3) where 3sσY is its volatility and 1.ssaY bYCRä Thus the wealth process is given by   3311dddd,TTTs ssttTsss sstTss stXxaYsσYWrY XμYrYπsσYW  (4) where x is the initial wealth, ,sstsTπF is the port- folio process with 221dTsstEσYπs. The investor’s objective is to maximize the expected M. ALGHALITH Copyright © 2011 SciRes. APM 82 utility of the terminal wealth ,, sup,tTtVtxyEuX F (5) where ().V is the value function, .u is a differentiable, bounded and concave utility function. Under regularity conditions, the value function is differentiable and thus satisfies the Hamiltonian-Jacobi- Bellman PDE     2222323 3221131312 121122sup120,ttxyyyxy xxtxttxxtxyVryxayVbyVσyV ρσ yσyV σyVyryVyyyVyyV     ,, ,VTxyux (6) where ijρ is the correlation coefficient between the Brownian motions. Hence, the optimal solution is    12 1221113 13.xxytxxμyryVρσ yσyVπσyVρσ yσy (7) Similar to the previous literature, an explicit solution can be obtained for specific forms of utility such as a logarithmic utility function. 3. Viscosity Solutions We can apply the constrained viscosity solutions to (6), given the HJB is degenerate elliptic and monotone increasing in V (see, for example, ). Consider this HJB   ,, ,0,Ω,,Ω,xxxHxVxV xVxxVxgx xää (8) where Ω is a bounded open set. Definition 1 A continuous function Vx is a viscosity subsolution of (6) if ,,,0, ,,HxV xPXPDV xXJVx xä滗 (9) A continuous function Vx is a viscosity supersolution of (8) if ,,,0, ,,,HxV xPXPDV xXJVxxä滗 (10) where   ,:limsup0 ,yxVyVxPy xDV xPyx(11)   ,: liminf0,yxVyVxPy xDV xPyx(12) are the super-differential and sub-differential, respectively; and  2,:1,,2lim sup0,yxJVx PXVy VxPyxXyxyxyx (13)  212,:,,lim inf0,yxJVx PXVyVxPy xXy xy xyx (14) are the superject and subject, respectively. A function Vx is a viscosity solution if it is both a viscosity subsolution and a viscosity supersolution. Proposition 1 Vx is the unique constrained viscosity solution of (6). Proof Let ΩVCä and let sV and iV be the upper and lower semicontinuous envelopes of V, respectively, where 12sup :,sVuxuuu 12inf :,iVu xuuu where u1 and u2 are sub-solution and super-solution, respectively. Thus ΩsV USCä and ΩiV LSCä are a viscosity subsolution and supersolution, respectively. At the boundary we have ,Vx sViV by the comparison principle  in Ω.sV iV By definition sViV and M. ALGHALITH Copyright © 2011 SciRes. APM 83thus  in ΩVx sViV is the unique viscosity solution. 4. References  M. Alghalith, “A New Stochastic Factor Model: General Explicit Solutions,” Applied Mathematics Letters, Vol. 22, No. 12, 2009, pp. 1852-1854. doi:10.1016/j.aml.2009.07.011  N. Castaneda-Leyva and D. Hernandez-Hernandez, “Op- timal Consumption-Investment Problems in Incomplete Markets with Random Coefficients,” Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, Sevilla, 12-15 December 2005, pp. 6650-6655.  J. Cvitanic and F. Zapatero, “Introduction to the Economics and Mathematics of Financial Markets,” MIT Press, Cambridge, 2004.  W. Fleming, “Some Optimal Investment, Production and Consumption Models,” Contemporary Mathematics, Vol. 351, 2004, pp. 115-123.  F. Focardi and F. Fabozzi, “The Mathematics of Financial Modeling and Investment Management,” Wiley E-Series, 2004.  F. Minani, “Hausdorff Continuous Viscosity Solutions to Hamilton-Jacobi Equations and their Numerical Analy-sis,” Unpublished Ph.D. Thesis, University of Pretoria, Pretoria, 2007.