Chebyshev polynomial approximation to approximate partial differential equations

6. Investment Under Uncertainty

The second empirical example we consider is taken from the Investment theory under
uncertainty. We consider the case of having to decide when it is optimal to shutdown a
machinery, assuming that there is no maintenance cost for maintaining the machine alive.
Suppose that π(t) is the profit generated by the machinery at time t , and suppose that it
follows the Brownian motion process below

dπ(t) = adt +σdz ,        π(0) = π₀             (7)

where a is the rate of depreciation of the investment, σis the volatility of profit, and
dz is an increment of a Wiener process.

We report three paths of the process in Figure 3. We have considered the parameters
reported at the bottom of Table 3 and the time T = 10 , has been divided in one-hundred
time steps. As it appears clear, the investment will, in general, produce a loss before 3
years. However the decision to shutdown the machinery cannot be only taken by looking
at the dynamics of these paths. In fact, this is a more complex problem since once the
machinery has been shutdown it cannot be re-started again. That is the investment is
irreversible. Therefore, one has to consider the optimal policy to decide when it is
convenient to shutdown the investment.

Following Dixit and Pindyck (1994), suppose that F(π,t)is a claim of the profit flow, π,
and suppose it is determined as

T

F(π, t) = max_τ∈Γ E∫ e ^-rtπ(t)dt                              (8)

0

where r is the risk free rate of interest, T is the time and τ ∈ Γis a random stopping time.

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