Consider an investor with initial wealth W0 > 0 who seeks to maximize E[logWT]. Assume E



T 0

|rt|dt



< ∞ and E



T 0

κ2 t dt



< ∞, where κ denotes the maximum Sharpe ratio. Assume portfolio processes are constrained to satisfy E



T 0

π

ttπt dt



< ∞.

Recall that this constraint implies E



T 0

π

σ dB



= 0 .

(a) Using the formula (13.15) for Wt show that the optimal portfolio process is

π = −1

(μ−rι).

Hint: The objective function obtained by substituting the formula

(13.15) for Wt can be maximized in π separately at each date and in each state of the world.

(b) Assume the market is Markovian. Show that the investor’s value function is V(t,w, x) = logw+ f(t,x), where f(t,x) = E



T t



rs +

1 2

κ 2 s



ds

#

#

#

#

Xt = x



.