, 1 Numerical Solution of Optimal Investment
Problems
Solving optimal investment problems is difficult manually and remains chal-
lenging with computers. Quick, precise solutions are harder to achieve than
often assumed, and universally reliable automated solvers for complex invest-
ments do not yet exist.
1.1 Discrete Time Framework
Numerical methods require discretization, so we turn to a discrete time frame-
work. Two main differences in notation between Delage et al. and Munk are
that portfolio weights are called x rather than π and the indirect utility
function is called V rather than J.
We denote by St a (possibly multi-dimensional) state process, by rt+∆t the
vector of excess returns of the risky assets over [t, t + ∆t], and by Rf the
(constant) risk-free return over [t, t + ∆t], i.e., Rf = erf ∆t or Rf = 1 + rf ∆t.
For portfolio weights xt invested in the risky assets, this results in the wealth
dynamics
Wt+∆t = Wt (Rf + x′t rt+∆t ).
We abstract from labor income and consumption (y = c = 0) and thus do not
need to care about time discounting either. This gives us the maximization
problem
V0 (W0 , S0 ) = max E0 [u(WT )]
(xt )∈A
under the constraint
Wt+∆t = Wt (Rf + x′t rt+∆t ).
Here A denotes the set of admissible portfolio strategies. Portfolio strategies
must be adapted, and possibly fulfill further constraints like short-selling or
borrowing.
Exercise 1 What do the short-selling and borrowing constraints look like as
conditions on x?
2
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