Numerical Analysis of a Free-Boundary Singular Control Problem in Financial Economics

Posted: 27 Sep 2005

See all articles by Ayman Hindy

Ayman Hindy


Steven H. Zhu

Banc of America Merrill Lynch; Morgan Stanley

Chi-fu Huang

Long Term Capital Management


We analyze a numerical scheme for solving the consumption and investment allocation problem studied in a companion paper by Hindy, Huang, and Zhu (1993). For this problem, Bellman's equation is a differential inequality involving second order partial differential equation with gradient constraints. The solution involves finding a free-boundary at which consumption occurs. We prove that the value function is the unique viscosity solution to Bellman's equation. We describe a numerical analysis technique based on Markov chain approximation schemes. We approximate the original continuous time problem by a sequence of discrete parameter Markov chains control problems. We prove that the value functions and the optimal investment policies in the sequence of the approximating control problems converge to the value function and the optimal investment policy, if it exists, of the original problem. We also show that the optimal consumption and abstinence regions in the sequence of approximating problems converge to those in the original problem.

JEL Classification: G00

Suggested Citation

Hindy, Ayman and Zhu, Steven H. and Huang, Chi-fu, Numerical Analysis of a Free-Boundary Singular Control Problem in Financial Economics. Journal of Economic Dynamics and Control, Vol 21, issue 2-3, 1997 . Available at SSRN:

Ayman Hindy


No Address Available

Steven H. Zhu (Contact Author)

Banc of America Merrill Lynch ( email )

Bank of America Plaza
335 Madison Ave, 5th Floor
New York, NY 10017
United States
646-855-1853 (Phone)


Morgan Stanley ( email )

1585 Broadway
New York, NY 10036
United States

Chi-fu Huang

Long Term Capital Management

Greenwich, CT 06830
United States

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