PROCEEDINGS Volume 1 - ineag

subject to the static budget constraint
and the non-negativity constraints in (11). In the static problem, optimization is carried out over the consumptionbequest
policy. The portfolio choice is a residual decision. The choice is determined by the need to finance the
optimal consumption-bequest policy.
Theorem 3. (Ocone and Karatzas (1991), Detemple, Garcia and Rindisbacher (2003)). The optimal consumption
and bequest policies are
(20)
where the state price density in (17) and is the unique solution of the equation
The optimal portfolio is with
where , and are the absolute risk tolerance measures
and evaluated at optimal consumption and bequest. Moreover,
where is the Malliavin derivative process that satisfies the linear stochastic differential equation
and are the gradients of with respect
to .
The marginal cost of consumption is the state price density adjusted by a Lagrange multiplier so as to account for
the need to satisfy the static budget constraint. Thus, and the optimal consumption-bequest
policies (20) follow. The equation for is then dictated by the need to saturate the budget constraint. Optimal wealth
is the present value of future consumption and bequest
The optimal portfolio determines the volatility of wealth. To identify it, it suffices to calculate the volatility
coefficients on the left and right hand sides of (26). This can be done by applying the Clark-Ocone formula (see
Ocone and Karatzas (1991) and Detemple, Garcia and Rindisbacher (2003)) and the rules of Malliavin calculus (see
Nualart (1995) for a comprehensive treatment). The leads to (22)-(23).
The portfolio component (22) corresponds to the mean-variance term (13). The new element is that the leading term,
which was written as a ratio of derivatives of the value function in the previous formula, is now seen to represent the
cost of optimal risk tolerance. The portfolio component (23) is the dynamic hedge due to fluctuations in the state
variables. This dynamic hedge has two fundamental parts. The first is motivated by fluctuations in the interest rate.
This is the term depending on in (24). It represents an interest rate hedge. The second is due to fluctuations
in the market prices of risk. It comprises the terms in in (24) and represents a market price of risk hedge.
The two hedges can be written as
20
(19)
(21)
(22)
(23)
(24)
(25)
(26)