It would be interesting to use the data prior to, say 1975, to see the result. I don't have data prior to 1965, but for Oil, Coal and Natural Gas they likely exist.
Do you want to dig deeper into this? We could try to find that data. Vencit omnia veritas.
dX1 = a X2 dt dX2 = -a X1 dt
describes a circle of constant radius, the stochastic (Ito) differential equation
implies
d(X1^2 + X2^2) = 2X1 dX1 + 2 X2 dX2 + dX1 dX1 + dX2 dX2 = a^2 (X1^2 + X2^2) dt
or
d(R^2) = a^2 R^2 dt
that is, R^2 grows exponentially with time in the stochastic case, even though it is constant in the ordinary case.
So, the secular 2% increase in total energy use might be "explained" by interpreting the extended Fisher - Pry - Marchetti model as a stochastic (Ito) differential equation. Can the last politician to go out the revolving door please turn the lights off?
the stochastic (Ito) differential equation dX1 = a X2 dt dX2 = -a X1 dt implies d(X1^2 + X2^2) = a^2 (X1^2 + X2^2) dt
d(X1^2 + X2^2) = a^2 (X1^2 + X2^2) dt
dX1 = X2 (a dt + s dz1) dX2 = X1(- a dt + s dz2) implies d(X1^2 + X2^2) = a^2 s^2 (X1^2 + X2^2) dt + X1 X2(s1 dz1 + s2 dz2)
d(X1^2 + X2^2) = a^2 s^2 (X1^2 + X2^2) dt + X1 X2(s1 dz1 + s2 dz2)
