This chapter is concerned with the consumption and portfolio selection problems formulated by R. C. Merton [112] in mathematical finance. We consider the optimization problems to maximize the expected utility function with respect to consumption rates c(t) and portfolio processes π(t) for the vector process S(t) of prices of N risky assets S1(t), …, SN(t) and general utility functions U(c). We try to find a connection between the HJB equations and the linear differential equations by using the Legendre transform. We study the existence of the smooth solutions of the HJB equations from the point of view of viscosity solutions.

The Model

Consider the consumption and portfolio selection problem. Define the following quantities:

1. S0(t) = price of the riskless asset at time t.

2. S(t) = vector process of prices of the N risky assets S1(t), …, SN(t) at time t.

3. B(t) = the N-dimensional standard Brownian motion.

4. X(t) = total wealth at time t.

5. π(t) = fraction of wealth in the risky assets at time t.

6. r = return on the riskless asset.

7. b = vector of expected return on the N risky assets.

8. σ = volatility N × N matrix of risky assets, σ ≠ 0.

9. c(t) = consumption rate at time t.

10. U(c) = the utility function.