K-medoids

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. 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 $p_t(x)$); we want to know the probability distribution of the state at a later time $s>t$. The adjective 'forward' refers to the fact that $p_t(x)$ serves as the initial condition and the PDE is integrated forward in time. (In the common case where the initial state is known exactly $p_t(x)$ 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 $u_s(x)$ which is equal to 1 if state x is in the target set at time s, and zero otherwise. In other words, $u_s(x)=1_B$, the indicator function for the set B. We want to know for every state x at time $t,(t<s)$ what is the probability of ending up in the target set at time s (sometimes called the hit probability). In this case $u_s(x)$ 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 $x(t)$ evolves according to the stochastic differential equation

$dx(t)=\mu (x(t),t)dt+\sigma (x(t),t)dW(t)$

then the Kolmogorov backward equation is as follows

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

for $t\le s$, subject to the final condition $p(x,s)=u_s(x)$. This can be derived using Itō's lemma on $p(x,t)$ and setting the 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 $u_s(x)$ over all paths that originate from state x at time t:

$P(X_s\in B\mid X_t=x)=E[u_s(x)\mid X_t=x]$

Historically of course the KBE 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:

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

for $s\ge t$, with initial condition $p(x,t)=p_t(x)$. For more on this equation see Fokker–Planck equation.

References

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