📂Probability Theory

Kolmogorov Differential Equation Derivation

Theorem

A differential equation holds for the transition probability matrix $P(t)$ and the differential matrix $Q$. $${{ d P(t) } \over { dt }} = Q P(t) = P(t) Q$$

Explanation

If one were to make a distinction, $dP/dt = P(t) Q$ is referred to as the backward Kolmogorov differential equation, and $dP/dt = Q P(t)$ as the forward Kolmogorov differential equation or Stochastic governing equation.

Derivation

According to the Chapman-Kolmogorov equation $P (t+h) = P (t) P (h)$ for continuous stochastic processes, the following holds. $$\begin{align*} {{ d P(t) } \over { dt }} =& \lim_{h \to 0} h^{-1} \left[ P (t+h) - P(t) \right] \\ =& \lim_{h \to 0} h^{-1} \left[ P (t) P (h) - P(t) \right] \\ =& P (t) \lim_{h \to 0} h^{-1} \left( P(h) - I \right) \\ =& P (t) Q \end{align*}$$ Here, $I$ is the identity matrix, and since $t + h = h + t$, by the same method, it can be shown that $P’(t) = Q P(t)$ also holds.

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See also

• This can also be considered as the Fokker-Planck equation version of the continuous Markov chain.