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 a 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.
-
Faculty
- Academic Areas
- Awards & Honors
- Seminars
-
Conferences
- Accounting Summer Camp
- California Econometrics Conference
- California Quantitative Marketing PhD Conference
- California School Conference
- China India Insights Conference
- Homo economicus, Evolving
-
Initiative on Business and Environmental Sustainability
- Political Economics (2023–24)
- Scaling Geologic Storage of CO2 (2023–24)
- A Resilient Pacific: Building Connections, Envisioning Solutions
- Adaptation and Innovation
- Changing Climate
- Civil Society
- Climate Impact Summit
- Climate Science
- Corporate Carbon Disclosures
- Earth’s Seafloor
- Environmental Justice
- Finance
- Marketing
- Operations and Information Technology
- Organizations
- Sustainability Reporting and Control
- Taking the Pulse of the Planet
- Urban Infrastructure
- Watershed Restoration
- Junior Faculty Workshop on Financial Regulation and Banking
- Ken Singleton Celebration
- Marketing Camp
- Quantitative Marketing PhD Alumni Conference
- Theory and Inference in Accounting Research
- Voices
- Publications
- Books
- Working Papers
- Case Studies
- Postdoctoral Scholars
-
Research Labs & Initiatives
- Cities, Housing & Society Lab
- Corporate Governance Research Initiative
- Corporations and Society Initiative
- Golub Capital Social Impact Lab
- Initiative for Financial Decision-Making
- Policy and Innovation Initiative
- Rapid Decarbonization Initiative
- Stanford Latino Entrepreneurship Initiative
- Value Chain Innovation Initiative
- Venture Capital Initiative
- Behavioral Lab
- Data, Analytics & Research Computing