[TECH] Exponential function (e-μt) is the only unique solution satifying R(x+t) = R(x)R(t)(semi-group property).

Exponential function keeps popping up every where stochastic analysis. However it has a crucial property which is often exploited during the analysis. The property I'm referring to is R(X+Y) = R(Y)R(Y) (e.g. X and Y) could be random variables). Now I prove a strong statement " R(X+Y) = R(X)R(Y) If and Only If R(x) = e^-μt where μ is some constant. The reverse direction (<==) is trivial because we can easily verify the fact by substituting R(x) with e^-μt. In the following we prove that other direction (==>).

One may wonder how about function a^x which also satisfies the semi-group property by inspection. However the μ takes care of that observe that a^x = e^ln(a)x.