Chebyshev polynomial approximation to approximate partial differential equations

Suppose that the machinery can be eliminated only up to time T . After that time it has to
run forever. We assume that if the machinery is eliminated cannot be re-installed, this
assumption highlights the irreversibility of the investment.

The Bellman Equation for the optimal stopping problem can be written as

F(π,t) = max(0,πdt + (1 - rdt)^-1E[F(π+ dπ,t + dt)]

Therefore, if the machinery is eliminated the profit will be zero. If we continue to hold
alive the machinery, then the profit is given by the conditional expectation in (9). One
can show that for this case, the value function F , in the continuation region, satisfies the
following Bellman equation

rF = π + aF_π +1/ 2σ² F_ππ

where F_(.) is the derivative with respect to the sub-script.

Under the assumption that if not abandoned by T , the machinery has to run forever, the

terminal condition can be written as F(π, T) = max(0, a / r^2 + r / π).

We can still use the same approach as in Section (3) to solve this optimal stopping
problem. However, now solving (10) at each stopping times require more effort. One way
could consist in using finite difference methods. However, this approach turns out to be
very time consuming. On the other hand using Richardson extrapolation methods would
increase the speed but at a cost of, sometimes, poor convergence. The method described
in Section (3) can be adapted to this specific case. In fact, it is very similar in spirit to the

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