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)
