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+---
+title: "Kolmogorov equations"
+firstLetter: "K"
+publishDate: 2021-11-14
+categories:
+- Mathematics
+- Statistics
+
+date: 2021-11-13T21:05:30+01:00
+draft: false
+markup: pandoc
+---
+
+# Kolmogorov equations
+
+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 $g$:
+
+$$\begin{aligned}
+    \dd{Y_s}
+    &= \bigg( \pdv{k}{s} + \pdv{k}{x} f + \frac{1}{2} \pdv[2]{k}{x} g^2 \bigg) \dd{s} + \pdv{k}{x} g \dd{B_s}
+    \\
+    &= \bigg( \pdv{k}{s} + \hat{L} k \bigg) \dd{s} + \pdv{k}{x} g \dd{B_s}
+\end{aligned}$$
+
+Where we have defined the linear operator $\hat{L}$
+to have the following action on $k$:
+
+$$\begin{aligned}
+    \hat{L} k
+    \equiv \pdv{k}{x} f + \frac{1}{2} \pdv[2]{k}{x} g^2
+\end{aligned}$$
+
+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**:
+
+$$\begin{aligned}
+    \boxed{
+        - \pdv{\psi}{s}
+        = \hat{L} \psi
+        = f \pdv{\psi}{x} + \frac{1}{2} g^2 \pdv[2]{\psi}{x}
+    }
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \mathbf{E}[Y_t]
+    = \mathbf{E}[k(X_t, t)]
+    = \int_{-\infty}^\infty k(y, t) \: \phi(t, y) \dd{y}
+    = \braket{k}{\phi}
+    = \mathrm{const}
+\end{aligned}$$
+
+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 $\pdv*{k}{t} + \hat{L} k = 0$:
+
+$$\begin{aligned}
+    0
+    = \pdv{t} \braket{k}{\phi}
+    = \braket{k}{\pdv{\phi}{t}} + \braket{\pdv{k}{t}}{\phi}
+    = \braket{k}{\pdv{\phi}{t}} - \braket{\hat{L} k}{\phi}
+    = \braket{k}{\pdv{\phi}{t} - \hat{L}{}^\dagger \phi}
+\end{aligned}$$
+
+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 $|y| \to \infty$:
+
+$$\begin{aligned}
+    \braket{\hat{L} k}{\phi}
+    &= \int_{-\infty}^\infty \pdv{k}{y} f \phi + \frac{1}{2} \pdv[2]{k}{y} g^2 \phi \dd{y}
+    \\
+    &= \bigg[ k f \phi + \frac{1}{2} \pdv{k}{y} g^2 \phi \bigg]_{-\infty}^\infty
+    - \int_{-\infty}^\infty k \pdv{y}(f \phi) + \frac{1}{2} \pdv{k}{y} \pdv{y}(g^2 \phi) \dd{y}
+    \\
+    &= \bigg[ -\frac{1}{2} k g^2 \phi \bigg]_{-\infty}^\infty
+    + \int_{-\infty}^\infty - k \pdv{y}(f \phi) + \frac{1}{2} k \pdv[2]{y}(g^2 \phi) \dd{y}
+    \\
+    &= \int_{-\infty}^\infty k \: \big( \hat{L}{}^\dagger \phi \big) \dd{y}
+    = \braket{k}{\hat{L}{}^\dagger \phi}
+\end{aligned}$$
+
+Since $k$ is arbitrary, and $\pdv*{\braket{k}{\phi}}{t} = 0$ for all $k$,
+we thus arrive at the **forward Kolmogorov equation**,
+describing the evolution of the probability density $\phi(t, y)$:
+
+$$\begin{aligned}
+    \boxed{
+        \pdv{\phi}{t}
+        = \hat{L}{}^\dagger \phi
+        = - \pdv{y}(f \phi) + \frac{1}{2} \pdv[2]{y}(g^2 \phi)
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  U.H. Thygesen,
+    *Lecture notes on diffusions and stochastic differential equations*,
+    2021, Polyteknisk Kompendie.