The Kolmogorov backward equation (KBE) (diffusion) and its adjoint sometimes known as the Kolmogorov forward equation (diffusion) are partial differential equations (PDE) that arise in the theory of continuous-time continuous-state Markov processes. Both were published by Andrey Kolmogorov in 1931.^[1] Later it was realized that the forward equation was already known to physicists under the name Fokker–Planck equation; the KBE on the other hand was new.

Informally, the Kolmogorov forward equation addresses the following problem. We have information about the state x of the system at time t (namely a probability distribution ${\displaystyle p_{t}(x)}$); we want to know the probability distribution of the state at a later time ${\displaystyle s>t}$. The adjective 'forward' refers to the fact that ${\displaystyle p_{t}(x)}$ serves as the initial condition and the PDE is integrated forward in time (in the common case where the initial state is known exactly, ${\displaystyle p_{t}(x)}$ is a Dirac delta function centered on the known initial state).

The Kolmogorov backward equation on the other hand is useful when we are interested at time t in whether at a future time s the system will be in a given subset of states B, sometimes called the target set. The target is described by a given function ${\displaystyle u_{s}(x)}$ which is equal to 1 if state x is in the target set at time s, and zero otherwise. In other words, ${\displaystyle u_{s}(x)=1_{B}}$, the indicator function for the set B. We want to know for every state x at time ${\displaystyle t,\ (t<s)}$ what is the probability of ending up in the target set at time s (sometimes called the hit probability). In this case ${\displaystyle u_{s}(x)}$ serves as the final condition of the PDE, which is integrated backward in time, from s to t.

Formulating the Kolmogorov backward equation

Assume that the system state ${\displaystyle X_{t}}$ evolves according to the stochastic differential equation

${\displaystyle dX_{t}=\mu (X_{t},t)\,dt+\sigma (X_{t},t)\,dW_{t}\,,}$

then the Kolmogorov backward equation is^[2]

${\displaystyle {\frac {\partial }{\partial t}}p(x,t)=\mu (x,t){\frac {\partial }{\partial x}}p(x,t)+{\frac {1}{2}}\sigma ^{2}(x,t){\frac {\partial ^{2}}{\partial x^{2}}}p(x,t),}$

for ${\displaystyle t\leq s}$, subject to the final condition ${\displaystyle p(x,s)=u_{s}(x)}$. This can be derived using Itō's lemma on ${\displaystyle p(x,t)}$ and setting the ${\displaystyle dt}$ term equal to zero.

This equation can also be derived from the Feynman–Kac formula by noting that the hit probability is the same as the expected value of ${\displaystyle u_{s}(x)}$ over all paths that originate from state ${\displaystyle x}$ at time ${\displaystyle t}$:

${\displaystyle \Pr(X_{s}\in B\mid X_{t}=x)=E[u_{s}(x)\mid X_{t}=x].}$

Historically, the KBE^[1] was developed before the Feynman–Kac formula (1949).

Formulating the Kolmogorov forward equation

With the same notation as before, the corresponding Kolmogorov forward equation is

${\displaystyle {\frac {\partial }{\partial s}}p(x,s)=-{\frac {\partial }{\partial x}}[\mu (x,s)p(x,s)]+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}[\sigma ^{2}(x,s)p(x,s)],}$

for ${\displaystyle s\geq t}$, with initial condition ${\displaystyle p(x,t)=p_{t}(x)}$. For more on this equation see Fokker–Planck equation.

See also

• Kolmogorov equations

References

• Etheridge, A. (2002). A Course in Financial Calculus. Cambridge University Press.