Problem Statement
The investor lives from time 0 to time T; his wealth at time t is denoted Wt. He starts with a known initial wealth W0 (which may include the present value of wage income). At time t he must choose what amount of his wealth to consume: ct and what fraction of wealth to invest in a stock portfolio: πt (the remaining fraction 1 − πt being invested in the risk-free asset).
The objective is
where E is the expectation operator, u is a known utility function (which applies both to consumption and to the terminal wealth, or bequest, WT) and δ is the subjective discount rate.
The wealth evolves according to the stochastic differential equation
where r is the risk-free rate, (μ, σ) are the expected return and volatility of the stock market and dBt is the increment of the Wiener process, i.e. the stochastic term of the SDE.
Additional assumptions. The utility function is of the constant relative risk aversion (CRRA) form:
Consumption cannot be negative: ct ≥ 0, while πt is unrestricted (that is borrowing or shorting stocks is allowed).
Investment opportunities are assumed constant, that is r, μ, σ are known and constant, in this (1969) version of the model, although Merton later allowed them to change.
Read more about this topic: Merton's Portfolio Problem
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