---
title: "Kolmogorov equations"
date: 2021-11-14
categories:
- Mathematics
- Statistics
- Stochastic analysis
layout: "concept"
---

Consider the following general [Itō diffusion](/know/concept/ito-calculus/)
$X_t \in \mathbb{R}$, which is assumed to satisfy
the conditions for unique existence on the entire time axis:

$$\begin{aligned}
    \dd{X}_t
    = f(X_t, t) \dd{t} + g(X_t, t) \dd{B_t}
\end{aligned}$$

Let $\mathcal{F}_t$ be the filtration to which $X_t$ is adapted,
then we define $Y_s$ as shown below,
namely as the [conditional expectation](/know/concept/conditional-expectation/)
of $h(X_t)$, for an arbitrary bounded function $h(x)$,
given the information $\mathcal{F}_s$ available at time $s \le t$.
Because $X_t$ is a [Markov process](/know/concept/markov-process/),
$Y_s$ must be $X_s$-measurable,
so it is a function $k$ of $X_s$ and $s$:

$$\begin{aligned}
    Y_s
    \equiv \mathbf{E}[h(X_t) | \mathcal{F}_s]
    = \mathbf{E}[h(X_t) | X_s]
    = k(X_s, s)
\end{aligned}$$

Consequently, we can apply Itō's lemma to find $\dd{Y_s}$
in terms of $k$, $f$ and $g$:

$$\begin{aligned}
    \dd{Y_s}
    &= \bigg( \pdv{k}{s} + \pdv{k}{x} f + \frac{1}{2} \pdvn{2}{k}{x} g^2 \bigg) \dd{s} + \pdv{k}{x} g \dd{B_s}
    \\
    &= \bigg( \pdv{k}{s} + \hat{L} k \bigg) \dd{s} + \pdv{k}{x} g \dd{B_s}
\end{aligned}$$

Where we have defined the linear operator $\hat{L}$
to have the following action on $k$:

$$\begin{aligned}
    \hat{L} k
    \equiv \pdv{k}{x} f + \frac{1}{2} \pdvn{2}{k}{x} g^2
\end{aligned}$$

At this point, we need to realize that $Y_s$ is
a [martingale](/know/concept/martingale/) with respect to $\mathcal{F}_s$,
since $Y_s$ is $\mathcal{F}_s$-adapted and finite,
and it satisfies the martingale property,
for $r \le s \le t$:

$$\begin{aligned}
    \mathbf{E}[Y_s | \mathcal{F}_r]
    = \mathbf{E}\Big[ \mathbf{E}[h(X_t) | \mathcal{F}_s] \Big| \mathcal{F}_r \Big]
    = \mathbf{E}\big[ h(X_t) \big| \mathcal{F}_r \big]
    = Y_r
\end{aligned}$$

Where we used the tower property of conditional expectations,
because $\mathcal{F}_r \subset \mathcal{F}_s$.
However, an Itō diffusion can only be a martingale
if its drift term (the one containing $\dd{s}$) vanishes,
so, looking at $\dd{Y_s}$, we must demand that:

$$\begin{aligned}
    \pdv{k}{s} + \hat{L} k
    = 0
\end{aligned}$$

Because $k(X_s, s)$ is a Markov process,
we can write it with a transition density $p(s, X_s; t, X_t)$,
where in this case $s$ and $X_s$ are given initial conditions,
$t$ is a parameter, and the terminal state $X_t$ is a random variable.
We thus have:

$$\begin{aligned}
    k(x, s)
    = \int_{-\infty}^\infty p(s, x; t, y) \: h(y) \dd{y}
\end{aligned}$$

We insert this into the equation that we just derived for $k$, yielding:

$$\begin{aligned}
    0
    = \int_{-\infty}^\infty \!\! \Big( \pdv{}{s}p(s, x; t, y) + \hat{L} p(s, x; t, y) \Big) h(y) \dd{y}
\end{aligned}$$

Because $h$ is arbitrary, and this must be satisfied for all $h$,
the transition density $p$ fulfills:

$$\begin{aligned}
    0
    = \pdv{}{s}p(s, x; t, y) + \hat{L} p(s, x; t, y)
\end{aligned}$$

Here, $t$ is a known parameter and $y$ is a "known" integration variable,
leaving only $s$ and $x$ as free variables for us to choose.
We therefore define the **likelihood function** $\psi(s, x)$,
which gives the likelihood of an initial condition $(s, x)$
given that the terminal condition is $(t, y)$:

$$\begin{aligned}
    \boxed{
        \psi(s, x)
        \equiv p(s, x; t, y)
    }
\end{aligned}$$

And from the above derivation,
we conclude that $\psi$ satisfies the following PDE,
known as the **backward Kolmogorov equation**:

$$\begin{aligned}
    \boxed{
        - \pdv{\psi}{s}
        = \hat{L} \psi
        = f \pdv{\psi}{x} + \frac{1}{2} g^2 \pdvn{2}{\psi}{x}
    }
\end{aligned}$$

Moving on, we can define the traditional
**probability density function** $\phi(t, y)$ from the transition density $p$,
by fixing the initial $(s, x)$
and leaving the terminal $(t, y)$ free:

$$\begin{aligned}
    \boxed{
        \phi(t, y)
        \equiv p(s, x; t, y)
    }
\end{aligned}$$

With this in mind, for $(s, x) = (0, X_0)$,
the unconditional expectation $\mathbf{E}[Y_t]$
(i.e. the conditional expectation without information)
will be constant in time, because $Y_t$ is a martingale:

$$\begin{aligned}
    \mathbf{E}[Y_t]
    = \mathbf{E}[k(X_t, t)]
    = \int_{-\infty}^\infty k(y, t) \: \phi(t, y) \dd{y}
    = \Inprod{k}{\phi}
    = \mathrm{const}
\end{aligned}$$

This integral has the form of an inner product,
so we switch to [Dirac notation](/know/concept/dirac-notation/).
We differentiate with respect to $t$,
and use the backward equation $\ipdv{k}{t} + \hat{L} k = 0$:

$$\begin{aligned}
    0
    = \pdv{}{t}\Inprod{k}{\phi}
    = \Inprod{k}{\pdv{\phi}{t}} + \Inprod{\pdv{k}{t}}{\phi}
    = \Inprod{k}{\pdv{\phi}{t}} - \Inprod{\hat{L} k}{\phi}
    = \Inprod{k}{\pdv{\phi}{t} - \hat{L}{}^\dagger \phi}
\end{aligned}$$

Where $\hat{L}{}^\dagger$ is by definition the adjoint operator of $\hat{L}$,
which we calculate using partial integration,
where all boundary terms vanish thanks to the *existence* of $X_t$;
in other words, $X_t$ cannot reach infinity at any finite $t$,
so the integrand must decay to zero for $|y| \to \infty$:

$$\begin{aligned}
    \Inprod{\hat{L} k}{\phi}
    &= \int_{-\infty}^\infty \pdv{k}{y} f \phi + \frac{1}{2} \pdvn{2}{k}{y} g^2 \phi \dd{y}
    \\
    &= \bigg[ k f \phi + \frac{1}{2} \pdv{k}{y} g^2 \phi \bigg]_{-\infty}^\infty
    - \int_{-\infty}^\infty k \pdv{}{y}(f \phi) + \frac{1}{2} \pdv{k}{y} \pdv{}{y}(g^2 \phi) \dd{y}
    \\
    &= \bigg[ -\frac{1}{2} k g^2 \phi \bigg]_{-\infty}^\infty
    + \int_{-\infty}^\infty - k \pdv{}{y}(f \phi) + \frac{1}{2} k \pdvn{2}{}{y}(g^2 \phi) \dd{y}
    \\
    &= \int_{-\infty}^\infty k \: \big( \hat{L}{}^\dagger \phi \big) \dd{y}
    = \Inprod{k}{\hat{L}{}^\dagger \phi}
\end{aligned}$$

Since $k$ is arbitrary, and $\ipdv{\Inprod{k}{\phi}}{t} = 0$ for all $k$,
we thus arrive at the **forward Kolmogorov equation**,
describing the evolution of the probability density $\phi(t, y)$:

$$\begin{aligned}
    \boxed{
        \pdv{\phi}{t}
        = \hat{L}{}^\dagger \phi
        = - \pdv{}{y}(f \phi) + \frac{1}{2} \pdvn{2}{}{y}(g^2 \phi)
    }
\end{aligned}$$

This can be rewritten in a way
that highlights the connection between Itō diffusions and physical diffusion,
if we define the **diffusivity** $D$, **advection** $u$, and **probability flux** $J$:

$$\begin{aligned}
    D
    \equiv \frac{1}{2} g^2
    \qquad \quad
    u
    = f - \pdv{D}{x}
    \qquad \quad
    J
    \equiv u \phi - D \pdv{\phi}{x}
\end{aligned}$$

Such that the forward Kolmogorov equation takes the following **conservative form**,
so called because it looks like a physical continuity equation:

$$\begin{aligned}
    \boxed{
        \pdv{\phi}{t}
        = - \pdv{J}{x}
        = - \pdv{}{x}\Big( u \phi - D \pdv{\phi}{x} \Big)
    }
\end{aligned}$$

Note that if $u = 0$, then this reduces to
[Fick's second law](/know/concept/ficks-laws/).
The backward Kolmogorov equation can also be rewritten analogously,
although it is less noteworthy:

$$\begin{aligned}
    \boxed{
        - \pdv{\psi}{t}
        = u \pdv{\psi}{x} + \pdv{}{x}\Big( D \pdv{\psi}{x} \Big)
    }
\end{aligned}$$

Notice that the diffusivity term looks the same
in both the forward and backward equations;
we say that diffusion is self-adjoint.



## References
1.  U.H. Thygesen,
    *Lecture notes on diffusions and stochastic differential equations*,
    2021, Polyteknisk Kompendie.