Kolmogorov equations (Markov jump process)

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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 [math]\displaystyle{ P(x,s;y,t) }[/math], where [math]\displaystyle{ x, y \in \Omega }[/math] (the state space) and [math]\displaystyle{ t \gt s }[/math] are the final and initial time respectively.

The equations

For the case of a countable state space we put [math]\displaystyle{ i,j }[/math] in place of [math]\displaystyle{ x,y }[/math]. The Kolmogorov forward equations read

[math]\displaystyle{ \frac{\partial P_{ij}}{\partial t}(s;t) = \sum_k P_{ik}(s;t) A_{kj}(t) }[/math],

where [math]\displaystyle{ A(t) }[/math] is the transition rate matrix (also known as the generator matrix),

while the Kolmogorov backward equations are

[math]\displaystyle{ \frac{\partial P_{ij}}{\partial s}(s;t) = -\sum_k A_{ik}(s) P_{kj}(s;t) }[/math]

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

[math]\displaystyle{ A_{ij}(t) = \left[\frac{\partial P_{ij}}{\partial u}(t;u)\right]_{u=t}, \quad A_{jk}(t) \ge 0,\ j\ne k, \quad \sum_k A_{jk}(t) =0. }[/math]

Background

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

Relation with the generating function

Still in the discrete state case, letting [math]\displaystyle{ s=0 }[/math] and assuming that the system initially is found in state [math]\displaystyle{ i }[/math], the Kolmogorov forward equations describe an initial-value problem for finding the probabilities of the process, given the quantities [math]\displaystyle{ A_{jk}(t) }[/math]. We write [math]\displaystyle{ p_k(t)= P_{ik}(0;t) }[/math] where [math]\displaystyle{ \sum_{k}p_k(t) = 1 }[/math], then

[math]\displaystyle{ \frac{d p_k}{dt}(t) = \sum_j A_{jk}(t) p_j(t);\quad p_k(0)=\delta_{ik}, \qquad k=0,1,\dots . }[/math]

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

[math]\displaystyle{ \Psi(x,t) = \sum_k x^k p_k(t),\quad }[/math]

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

[math]\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. }[/math]

History

A brief historical note can be found at Kolmogorov equations.

See also

• Continuous-time Markov process
• Jump process
• Master equation
• Fokker–Planck equation
• Kolmogorov backward equations (diffusion)

References

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