Q2. Why can the HJB equation be degenerate?
Because of market incompleteness, in evidence in the stochastic income stream and the imperfect correlation of its source of noise with that of the stock price, the HJB equation can be degenerate and the value function therefore need not be smooth.
Q3. What is the value function for the original income-hedging problem?
The reduced state variable z for the original income-hedging problem can also be viewed as the wealth state variable for a new investment-consumption problem, in which the utility function is not HARA and in which a fixed fraction of wealth must be held in an asset whose returns are uncorrelated with the returns from the available risky security.
Q4. What is the main difficulty for the problem at hand?
The main difficulty for the problem at hand is that neither control, consumption rate nor risky investment, is uniformly bounded.
Q5. What is the first sign that u is smooth?
When y > 0, the authors first observe that the candidate value function 6(x, y) = y%+‘y) is smooth116 D. Dujie et al. / Journal of Economic Dynamics and Control 21 (1997) 753-782because u coincides with w, which is smooth.
Q6. What is the significance of the characterization of u?
It turns out that this characterization of u is crucial for proving regularity results for the value function v as well as for obtaining feedback forms for the optimal policies.
Q7. What is the value function of the stochastic control problem?
The value function of the stochastic control problem is a smooth solution of the associated Hamilton-Jacobi-Bellman (HJB) equation.
Q8. What is the expected rate of return of the risky asset?
The total expected rate of return of the risky asset is therefore b = a + 6.A consumption process is an element of the space LZ’+ consisting of any non-negative {p,}-progressively measurable process C such that E(si C,dt) < co for any T > 0.
Q9. What is the scale measure S associated with Z?
The scale measure S associated with Z is defined byS(z) = $4 dt,where45) = ew[ - ],:($$)dz], where x0 > 0 and to > 0 are arbitrary.
Q10. What is the way to show that the feedback policy is optimal?
In order to show that (C*, ZI*), as given by the feedback policy (g, h) is optimal, the authors first show that it exists and is admissible under the assumptions of the theorem, and then show that j(C*) = u(x, y).
Q11. What is the simplest way to determine if u is concave?
(3.4)After performing the (formal) maximization in (3.2) (assuming that u is smooth and strictly concave), the authors get: d2 bJ = ; f12z2(1 - p2)u” - & % + kzu’ + F(d) (z > O), (3.5)wherek=pk,a+k 8 2. (3.6)758 D. Duffie et al. /Journal of Economic Dynamics and Control 21 (1997) 753-782