The Dynamic Programming Equation for the Problem of Optimal Investment Under Capital Gains Taxes

This paper considers an extension of the Merton optimal investment problem to the case where the risky asset is subject to transaction costs and capital gains taxes. We derive the dynamic programming equation in the sense of constrained viscosity solutions. We next introduce a family of functions $(V_\varepsilon)_{\varepsilon>0}$, which converges to our value function uniformly on compact subsets, and which is characterized as the unique constrained viscosity solution of an approximation of our dynamic programming equation. In particular, this result justifies the numerical results reported in the accompanying paper [I. Ben Tahar, H. M. Soner, and N. Touzi (2005), Modeling Continuous-Time Financial Markets with Capital Gains Taxes, preprint, http://www.cmap.polytechnique.fr/$\sim$touzi/bst06.pdf].

[1]  R. C. Merton,et al.  Optimum consumption and portfolio rules in a continuous - time model Journal of Economic Theory 3 , 1971 .

[2]  H. Soner,et al.  Modeling continuous-time financial markets with capital gains taxes , 2007 .

[3]  Robert M. Dammon,et al.  Optimal Consumption and Investment with Capital Gains Taxes , 2001 .

[4]  M. James.  Controlled markov processes and viscosity solutions , 1994 .

[5]  G. Barles,et al.  A STRONG COMPARISON RESULT FOR THE BELLMAN EQUATION ARISING IN STOCHASTIC EXIT TIME CONTROL PROBLEMS AND ITS APPLICATIONS , 1998 .

[6]  H. Leland.,et al.  Optimal Portfolio Management with Transactions Costs and Capital Gains Taxes , 1999 .

[7]  R. C. Merton,et al.  Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case , 1969 .

[8]  H. Soner,et al.  Optimal Investment and Consumption with Transaction Costs , 1994 .

[9]  G. Kallianpur.  Stochastic differential equations and diffusion processes , 1981 .

[10]  Vivek S. Borkar,et al.  Optimal Control of Diffusion Processes , 1989 .

[11]  Optimal Investment with Taxes: An Existence Result , 1999 .

[12]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[13]  George M. Constantinides,et al.  Capital Market Equilibrium with Personal Tax , 1983 .

[14]  N. Karoui.  Les Aspects Probabilistes Du Controle Stochastique , 1981 .

[15]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[16]  R. C. Merton,et al.  Optimum Consumption and Portfolio Rules in a Continuous-Time Model* , 1975 .

[17]  G. Barles,et al.  The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems , 1995 .

[18]  H. Soner.  Optimal control with state-space constraint I , 1986 .

[19]  Ron Kaniel,et al.  Tax Management Strategies with Multiple Risky Assets , 2004 .

[20]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[21]  Raman Uppal,et al.  Portfolio Investment with the Exact Tax Basis Via Nonlinear Programming , 2004, Manag. Sci..

[22]  Optimal investment with taxes: an optimal control problem with endogeneous delay , 1999 .

[23]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[24]  J. Cox,et al.  Optimal consumption and portfolio policies when asset prices follow a diffusion process , 1989 .

[25]  M. Akian,et al.  On an Investment-Consumption Model With Transaction Costs , 1996 .