Initial commit after migration from Hugo

1 files changed, 246 insertions, 0 deletions

diff --git a/source/know/concept/kolmogorov-equations/index.md b/source/know/concept/kolmogorov-equations/index.md
new file mode 100644
index 0000000..a6579b2
--- /dev/null
+++ b/source/know/concept/kolmogorov-equations/index.md
@@ -0,0 +1,246 @@
+---
+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.