The Pdot-approximation is finding the shadow price of a stock, \(p\) from the relation:
\(p(s) = \frac{W_{s}(s) + \dot{p}(s)}{\delta - \dot{s}_{s}}\),
where \(W_{s} = \frac{dW}{ds}\), \( \dot{p}(s) = \frac{dp}{ds}\),
\(\dot{s}_{s} = \frac{d\dot{s}}{ds} \), and \(\delta\) is the given discount rate.
In order to operationalize this approach, we take the time derivative of this expression:
\( \dot{p} = \frac{ \left( \left(W_{ss}\dot{s} + \ddot{p} \right) \left( \delta - \dot{s}_{s} \right) +
\left( W_{s} + \dot{p} \right) \left(\dot{s}_{ss} \dot{s} \right) \right) }
{ \left( \delta - \dot{s}_{s} \right)^{2} } \)
Consider approximation \( \dot{p}(s) = \mathbf{\mu}(s)\mathbf{\beta}\), \(\mathbf{\mu}(s)\)
is Chebyshev polynomials and \(\mathbf{\beta}\) is their coeffcients.
Then, \( \ddot{p} = \frac{ d \dot{p}}{ds} \frac{ds}{dt} = diag (\dot{s}) \mathbf{\mu}_{s}(s) \mathbf{\beta}\) by the orthogonality of Chebyshev basis.
Adopting the properties above, we can get the unknown coefficient vector \(\beta\) from:
\( \mathbf{\mu \beta} = diag \left( \delta - \dot{s}_{s} \right)^{-2}
\left[ \left(W_{ss}\dot{s} + diag (\dot{s}) \mathbf{\mu}_{s} \mathbf{\beta} \right)\left( \delta - \dot{s}_{s} \right) +
diag \left(\dot{s}_{ss} \dot{s} \right) \left( W_{s} + \mathbf{\mu \beta} \right) \right] \), and
\(\mathbf{\beta} = \left[ diag \left( \delta - \dot{s}_{s} \right)^{2} \mathbf{\mu} - diag \left( \dot{s}\left( \delta - \dot{s}_{s} \right) \right) \mathbf{\mu}_{s}
- diag (\dot{s}_{ss} \dot{s} ) \mathbf{\mu} \right]^{-1}
\left( W_{ss} \dot{s} \left( \delta - \dot{s}_{s} \right) + W_{s} \dot{s}_{ss} \dot{s} \right) \).
If we suppose \( A = \left[ diag \left( \delta - \dot{s}_{s} \right)^{2} \mathbf{\mu} - diag \left( \dot{s}\left( \delta - \dot{s}_{s} \right) \right) \mathbf{\mu}_{s}
- diag (\dot{s}_{ss} \dot{s} ) \mathbf{\mu} \right] \) and
\( B = \left( W_{ss} \dot{s} \left( \delta - \dot{s}_{s} \right) + W_{s} \dot{s}_{ss} \dot{s} \right) \),
then over-determined case can be calculated:
\( \mathbf{\beta} = \left( A^{T}A \right)^{-1} A^{T}B \).
For more detils see Fenichel and Abbott (2014).