Assume (15.25) holds with strict inequality. Repeating the argument at the end of Section 15.5 shows that, for any date t, Et T

t MuCu du = Et T

t Mˆ uCˆ u du + XtEt T

t Mue−(a−b)(u−t)

du.

(Section 15.5 considers t = 0.) Assume power utility: u(c−x) = 1 1−ρ (c−x)1−ρ.

Assume the information in the economy is generated by a single Brownian motion B, there is a constant risk-free rate r, and there is a single risky asset with constant expected rate of return μ and constant volatility σ.

(a) Show that the optimal Cˆ is Cˆt = Ke−(δ/ρ)t Mˆ − 1

ρ

t for a constant K.

(b) Define

γ (t) = 1 r + a−b 1 −e−(r+a−b)(T−t)



.

Show that Et T

t Mu Mt e−(a−b)(u−t)

du = γ (t), Mˆ t = [1+ bγ (t)]Mt , 1 Mt Et T

t Mˆ uCˆ u du = β(t)M− 1

ρ

t , for a nonrandom function β.

(c) Define Wt = Et T

t Mu Mt Cu du.

Show that dWt = β(t)M− 1

ρ

t

μ− r

ρσ 

dBt +somethingdt

= [Wt −γ (t)Xt]

μ− r

ρσ 

dBt +somethingdt .

(d) Show that the optimal portfolio is

πt =



1− γ (t)Xt Wt

 μ− r

ρσ2 .