Module merton 

Merton’s log-utility portfolio problem.

A validating instance of ControlProblem with a known closed-form solution. The state is wealth x; the control u in [0, 1] is the fraction of wealth allocated to a risky asset.

Dynamics

dS/S = mu dt + sigma dW        (risky asset)
r     = risk-free rate          (riskless asset)
dx    = x [ r + u (mu - r) ] dt + x u sigma dW

Objective

Maximize expected log utility of terminal wealth:

J(u) = E[ ln x_T ]

HJB equation

0 = d_t V + sup_u { x (r + u (mu - r)) V_x
     + 0.5 (x u sigma)^2 V_xx }

Exact solution

The optimal portfolio fraction is constant:

u* = (mu - r) / sigma^2

and the value function is

V(t, x) = ln x + [ r + 0.5 (mu - r)^2 / sigma^2 ] (T - t)

where T - t is the remaining horizon.

Structs

Merton

Merton log-utility portfolio problem.