# Kolmogorov equations (Markov jump process)

In the context of a continuous-time Markov process, the Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, are a pair of systems of differential equations that describe the time-evolution of the probability ${\displaystyle P(x,s;y,t)}$, where ${\displaystyle x,y\in \Omega }$ (the state space) and ${\displaystyle t>s}$ are the final and initial time respectively.

## The equations

For the case of countable state space we put ${\displaystyle i,j}$ in place of ${\displaystyle x,y}$. Kolmogorov forward equations read

${\displaystyle {\frac {\partial P_{ij}}{\partial t}}(s;t)=\sum _{k}P_{ik}(s;t)A_{kj}(t)}$

while Kolmogorov backward equations are

${\displaystyle {\frac {\partial P_{ij}}{\partial s}}(s;t)=-\sum _{k}A_{ik}(s)P_{kj}(s;t)}$

The functions ${\displaystyle P_{ij}(s;t)}$ are continuous and differentiable in both time arguments. They represent the probability that the system that was in state ${\displaystyle i}$ at time ${\displaystyle s}$ jumps to state ${\displaystyle j}$ at some later time ${\displaystyle t>s}$. The continuous quantities ${\displaystyle A_{ij}(t)}$ satisfy

${\displaystyle A_{ij}(t)=\left[{\frac {\partial P_{ij}}{\partial u}}(t;u)\right]_{u=t},\quad A_{jk}(t)\geq 0,\ j\neq k,\quad \sum _{k}A_{jk}(t)=0.}$

## Background

The original derivation of the equations by Kolmogorov [1] starts with the Chapman-Kolmogorov equation (Kolmogorov called it Fundamental equation) for time-continuous and differentiable Markov processes on a finite, discrete state space. In this formulation, it is assumed that the probabilities ${\displaystyle P(i,s;j,t)}$ are continuous and differentiable functions of ${\displaystyle t>s}$. Also adequate limit properties for the derivatives are assumed. Feller [2] derives the equations under slightly different conditions, starting with the concept of purely discontinuous Markov process and formulating them for more general state spaces. Feller [2] proves the existence of solutions of probabilistic character to the Kolmogorov forward equations and Kolmogorov backward equations under natural conditions.

## Relation with the generating function

Still in the discrete state case, letting ${\displaystyle s=0}$ and assuming that the system initially is found in state ${\displaystyle i}$, The Kolmogorov forward equations describe an initial value problem for finding the probabilities of the process, given the quantities ${\displaystyle A_{jk}(t)}$. We put ${\displaystyle P_{ik}(0;t)=P_{k}(t)}$ and

${\displaystyle {\frac {dP_{k}}{dt}}(t)=\sum _{j}A_{jk}(t)P_{j}(t);\quad P_{k}(0)=\delta _{ik},\qquad k=0,1,\dots .}$

For the case of a pure death process with constant rates the only nonzero coefficients are ${\displaystyle A_{j,j-1}=\mu ,\ j\geq 1}$. Letting

${\displaystyle \Psi (x,t)=\sum _{k}x^{k}P_{k}(t),\quad }$

the system of equations can in this case be recast as a partial differential equation for ${\displaystyle {\Psi }(x,t)}$ with initial condition ${\displaystyle \Psi (x,0)=x^{i}}$. After some manipulations, the system of equations reads,[3]

${\displaystyle {\frac {\partial \Psi }{\partial t}}(x,t)=\mu (1-x){\frac {\partial {\Psi }}{\partial x}}(x,t);\qquad \Psi (x,0)=x^{i},\quad \Psi (1,t)=1.}$

## History

A brief historical note can be found at Kolmogorov equations