Expand knowledge base

15 files changed, 509 insertions, 1 deletions

diff --git a/content/know/category/measure-theory.md b/content/know/category/measure-theory.md
new file mode 100644
index 0000000..24ea18d
--- /dev/null
+++ b/content/know/category/measure-theory.md
@@ -0,0 +1,9 @@
+---
+title: "Measure theory"
+firstLetter: "M"
+date: 2021-11-26T10:21:09+01:00
+draft: false
+layout: "category"
+---
+
+This page will fill itself.
diff --git a/content/know/category/stochastic-analysis.md b/content/know/category/stochastic-analysis.md
new file mode 100644
index 0000000..ca82072
--- /dev/null
+++ b/content/know/category/stochastic-analysis.md
@@ -0,0 +1,9 @@
+---
+title: "Stochastic analysis"
+firstLetter: "S"
+date: 2021-11-26T10:13:21+01:00
+draft: false
+layout: "category"
+---
+
+This page will fill itself.
diff --git a/content/know/concept/conditional-expectation/index.pdc b/content/know/concept/conditional-expectation/index.pdc
index 5a8f07e..50120e8 100644
--- a/content/know/concept/conditional-expectation/index.pdc
+++ b/content/know/concept/conditional-expectation/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-10-23
 categories:
 - Mathematics
 - Statistics
+- Measure theory
 
 date: 2021-10-22T15:19:23+02:00
 draft: false
diff --git a/content/know/concept/detailed-balance/index.pdc b/content/know/concept/detailed-balance/index.pdc
new file mode 100644
index 0000000..1645c2d
--- /dev/null
+++ b/content/know/concept/detailed-balance/index.pdc
@@ -0,0 +1,238 @@
+---
+title: "Detailed balance"
+firstLetter: "D"
+publishDate: 2021-11-27
+categories:
+- Physics
+- Mathematics
+- Stochastic analysis
+
+date: 2021-11-25T20:42:35+01:00
+draft: false
+markup: pandoc
+---
+
+# Detailed balance
+
+Consider a system that can be regarded as a
+[Markov process](/know/concept/markov-process/),
+which means that its components (e.g. particles) are transitioning
+between a known set of states,
+with no history-dependence and no appreciable influence from interactions.
+
+At equilibrium, the principle of **detailed balance** then says that
+for all states, the rate of leaving that state is exactly equal to
+the rate of entering it, for every possible transition.
+In effect, such a system looks "frozen" to an outside observer,
+since all net transition rates are zero.
+
+We will focus on the case where both time and the state space are continuous.
+Given some initial conditions,
+assume that a component's trajectory can be described
+as an [Itō diffusion](/know/concept/ito-calculus/) $X_t$
+with a time-independent drift $f$ and intensity $g$,
+and with a probability density $\phi(t, x)$ governed by the
+[forward Kolmogorov equation](/know/concept/kolmogorov-equations/)
+(in 3D):
+
+$$\begin{aligned}
+    \pdv{\phi}{t}
+    = - \nabla \cdot \big( \vb{u} \phi - D \nabla \phi \big)
+\end{aligned}$$
+
+We start by demanding **stationarity**,
+which is a weaker condition than detailed balance.
+We want the probability $P$ of being in an arbitrary state volume $V$
+to be constant in time:
+
+$$\begin{aligned}
+    0
+    = \pdv{t} P(X_t \in V)
+    = \pdv{t} \int_V \phi \dd{V}
+    = \int_V \pdv{\phi}{t} \dd{V}
+\end{aligned}$$
+
+We substitute the forward Kolmogorov equation,
+and apply the divergence theorem:
+
+$$\begin{aligned}
+    0
+    = - \int_V \nabla \cdot \big( \vb{u} \phi - D \nabla \phi \big) \dd{V}
+    = - \oint_{\partial V} \big( \vb{u} \phi - D \nabla \phi \big) \cdot \dd{\vb{S}}
+\end{aligned}$$
+
+In other words, the "flow" of probability *into* the volume $V$
+is equal to the flow *out of* $V$.
+If such a probability density exists,
+it is called a **stationary distribution** $\phi(t, x) = \pi(x)$.
+Because $V$ was arbitrary, $\pi$ can be found by solving:
+
+$$\begin{aligned}
+    0
+    = - \nabla \cdot \big( \vb{u} \pi - D \nabla \pi \big)
+\end{aligned}$$
+
+Therefore, stationarity means that the state transition rates are constant.
+To get detailed balance, however, we demand that
+the transition rates are zero everywhere:
+the probability flux through an arbitrary surface $S$ must vanish
+(compare to closed surface integral above):
+
+$$\begin{aligned}
+    0
+    = - \int_{S} \big( \vb{u} \phi - D \nabla \phi \big) \cdot \dd{\vb{S}}
+\end{aligned}$$
+
+And since $S$ is arbitrary, this is only satisfied if the flux is trivially zero
+(the above justification can easily be repeated in 1D, 2D, 4D, etc.):
+
+$$\begin{aligned}
+    \boxed{
+        0 = \vb{u} \phi - D \nabla \phi
+    }
+\end{aligned}$$
+
+This is a stronger condition that stationarity,
+but fortunately often satisfied in practice.
+
+The fact that a system in detailed balance appears "frozen"
+implies it is **time-reversible**,
+meaning its statistics are the same for both directions of time.
+Formally, given two arbitrary functions $h(x)$ and $k(x)$,
+we have the property:
+
+$$\begin{aligned}
+    \boxed{
+        \mathbf{E}\big[ h(X_0) \: k(X_t) \big]
+        = \mathbf{E}\big[ h(X_t) \: k(X_0) \big]
+    }
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-reversibility"/>
+<label for="proof-reversibility">Proof</label>
+<div class="hidden">
+<label for="proof-reversibility">Proof.</label>
+Consider the following weighted inner product,
+whose weight function is a stationary distribution $\pi$
+satisfying detailed balance,
+where $\hat{L}$ is the Kolmogorov operator:
+
+$$\begin{aligned}
+    \braket*{\hat{L} h}{k}_\pi
+    \equiv \int_{-\infty}^\infty \hat{L}\{h(x)\} \: \pi(x) \: k(x) \dd{x}
+    = \int_{-\infty}^\infty h(x) \: \hat{L}{}^\dagger\{\pi(x) k(x)\} \dd{x}
+\end{aligned}$$
+
+Where we have used the definition of an adjoint operator.
+We would like to rewrite this:
+
+$$\begin{aligned}
+    \hat{L}{}^\dagger \{\pi k\}
+    = -\nabla \cdot \big( \vb{u} \pi k - D \nabla(\pi k) \big)
+    = -\nabla \cdot (\vb{u} \pi k - D k \nabla \pi - D \pi \nabla k)
+\end{aligned}$$
+
+Since $\pi$ is stationary by definition,
+we know that $\nabla \cdot (\vb{u} \pi - D \nabla \pi) = 0$,
+meaning:
+
+$$\begin{aligned}
+    \hat{L}{}^\dagger \{\pi k\}
+    = \nabla \cdot (D \pi \nabla k)
+    = \nabla \pi \cdot (D \nabla k) + \pi \nabla \cdot (D \nabla k)
+\end{aligned}$$
+
+Detailed balance demands that $\vb{u} \pi = D \nabla \pi$,
+leading to the following:
+
+$$\begin{aligned}
+    \hat{L}{}^\dagger \{\pi k\}
+    &= D \nabla \pi \cdot \nabla k + \pi \nabla \cdot (D \nabla k)
+    = \pi \vb{u} \cdot \nabla k + \pi \nabla \cdot (D \nabla k)
+    \\
+    &= \pi \big( \vb{u} \cdot \nabla k + \nabla \cdot (D \nabla k) \big)
+    = \pi \hat{L}\{k\}
+\end{aligned}$$
+
+Where we recognized the definition of $\hat{L}$
+from the backward Kolmogorov equation.
+Now that we have established that $\hat{L}{}^\dagger\{\pi k\} = \pi \hat{L}\{k\}$,
+we return to the inner product:
+
+$$\begin{aligned}
+    \braket*{\hat{L} h}{k}_\pi
+    = \int_{-\infty}^\infty h(x) \: \pi(x) \: \hat{L}\{k(x)\} \dd{x}
+    = \braket*{h}{\hat{L} k}_\pi
+\end{aligned}$$
+
+Consequently, the following weighted inner products must also be equivalent:
+
+$$\begin{aligned}
+    \braket{\exp\!(t \hat{L}) h}{k}_\pi
+    = \braket{h}{\exp\!(t \hat{L}) k}_\pi
+\end{aligned}$$
+
+Now, consider the time evolution of the
+[conditional expectation](/know/concept/conditional-expectation/)
+$\mathbf{E}\big[ k(X_t) | X_0 \big]$:
+
+$$\begin{aligned}
+    \pdv{t} \mathbf{E}\big[ k(X_t) | X_0 \big]
+    &= \pdv{t} \int_{-\infty}^\infty k(x) \: \phi(t, x) \dd{x}
+    = \int_{-\infty}^\infty k \pdv{\phi}{t} \dd{x}
+    \\
+    &= \int_{-\infty}^\infty k \: \hat{L}{}^\dagger\{\phi\} \dd{x}
+    = \int_{-\infty}^\infty \hat{L}\{k\} \: \phi \dd{x}
+    = \mathbf{E}\big[ \hat{L}\{k(X_t)\} | X_0 \big]
+\end{aligned}$$
+
+Where we used the forward Kolmogorov equation
+and the definition of an adjoint operator.
+Therefore, since the expectation $\mathbf{E}$
+does not explicitly depend on $t$ (only implicitly via $X_t$),
+we can naively move the differentiation inside
+(only valid within $\mathbf{E}$):
+
+$$\begin{aligned}
+    \pdv{t} \mathbf{E}\big[ k(X_t) | X_0 \big]
+    = \mathbf{E}\bigg[ \pdv{k(X_t)}{t} \bigg| X_0 \bigg]
+    = \mathbf{E}\bigg[ \hat{L}\{k(X_0)\} \bigg| X_0 \bigg]
+\end{aligned}$$
+
+A differential equation of the form $\pdv*{k}{t} = \hat{L}\{k(t, x)\}$
+for a time-independent operator $\hat{L}$
+has a general solution $k(t, x) = \exp\!(t \hat{L})\{k(0,x)\}$,
+therefore:
+
+$$\begin{aligned}
+    \mathbf{E}\big[ k(X_t) \big| X_0 \big]
+    = \mathbf{E}\big[ \exp\!(t \hat{L})\{k(X_0)\} \big| X_0 \big]
+    = \exp\!(t \hat{L})\{k(X_0)\}
+\end{aligned}$$
+
+With this, we can evaluate the two weighted inner products from earlier,
+which we know are equal to each other.
+Using the *tower property* of the conditional expectation:
+
+$$\begin{aligned}
+    \braket{h}{\exp\!(t \hat{L}) k}_\pi
+    &= \mathbf{E}\big[ h(X_0) \: \mathbf{E}[k(X_t) | X_0] \big]
+    = \mathbf{E}\big[ h(X_0) \: k(X_t) \big]
+    \\
+    = \braket{\exp\!(t \hat{L}) h}{k}_\pi
+    &= \mathbf{E}\big[ \mathbf{E}[h(X_t) | X_0] \: k(X_0) \big]
+    = \mathbf{E}\big[ h(X_t) \: k(X_0) \big]
+\end{aligned}$$
+
+Where the integral gave the expectation value at $X_0$,
+since $\pi$ does not change in time.
+</div>
+</div>
+
+
+
+## References
+1.  U.H. Thygesen,
+    *Lecture notes on diffusions and stochastic differential equations*,
+    2021, Polyteknisk Kompendie.
diff --git a/content/know/concept/dynkins-formula/index.pdc b/content/know/concept/dynkins-formula/index.pdc
new file mode 100644
index 0000000..a6aa2c4
--- /dev/null
+++ b/content/know/concept/dynkins-formula/index.pdc
@@ -0,0 +1,199 @@
+---
+title: "Dynkin's formula"
+firstLetter: "D"
+publishDate: 2021-11-28
+categories:
+- Mathematics
+- Stochastic analysis
+
+date: 2021-11-26T10:10:09+01:00
+draft: false
+markup: pandoc
+---
+
+# Dynkin's formula
+
+Given an [Itō diffusion](/know/concept/ito-calculus/) $X_t$
+with a time-independent drift $f$ and intensity $g$
+such that the diffusion uniquely exists on the $t$-axis.
+We define the **infinitesimal generator** $\hat{A}$
+as an operator with the following action on a given function $h(x)$,
+where $\mathbf{E}$ is a
+[conditional expectation](/know/concept/conditional-expectation/):
+
+$$\begin{aligned}
+    \boxed{
+        \hat{A}\{h(X_0)\}
+        \equiv \lim_{t \to 0^+} \bigg[ \frac{1}{t} \mathbf{E}\Big[ h(X_t) - h(X_0) \Big| X_0 \Big] \bigg]
+    }
+\end{aligned}$$
+
+Which only makes sense for $h$ where this limit exists.
+The assumption that $X_t$ does not have any explicit time-dependence
+means that $X_0$ need not be the true initial condition;
+it can also be the state $X_s$ at any $s$ infinitesimally smaller than $t$.
+
+Conveniently, for a sufficiently well-behaved $h$,
+the generator $\hat{A}$ is identical to the Kolmogorov operator $\hat{L}$
+found in the [backward Kolmogorov equation](/know/concept/kolmogorov-equations/):
+
+$$\begin{aligned}
+    \boxed{
+        \hat{A}\{h(x)\}
+        = \hat{L}\{h(x)\}
+    }
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-kolmogorov"/>
+<label for="proof-kolmogorov">Proof</label>
+<div class="hidden">
+<label for="proof-kolmogorov">Proof.</label>
+We define a new process $Y_t \equiv h(X_t)$, and then apply Itō's lemma, leading to:
+
+$$\begin{aligned}
+    \dd{Y_t}
+    &= \bigg( \pdv{h}{x} f(X_t) + \frac{1}{2} \pdv[2]{h}{x} g^2(X_t) \bigg) \dd{t} + \pdv{h}{x} g(X_t) \dd{B_t}
+    \\
+    &= \hat{L}\{h(X_t)\} \dd{t} + \pdv{h}{x} g(X_t) \dd{B_t}
+\end{aligned}$$
+
+Where we have recognized the definition of $\hat{L}$.
+Integrating the above equation yields:
+
+$$\begin{aligned}
+    Y_t
+    = Y_0 + \int_0^t \hat{L}\{h(X_s)\} \dd{s} + \int_0^\tau \pdv{h}{x} g(X_s) \dd{B_s}
+\end{aligned}$$
+
+As always, the latter [Itō integral](/know/concept/ito-integral/)
+is a [martingale](/know/concept/martingale/), so it vanishes
+when we take the expectation conditioned on the "initial" state $X_0$, leaving:
+
+$$\begin{aligned}
+    \mathbf{E}[Y_t | X_0]
+    = Y_0 + \mathbf{E}\bigg[ \int_0^t \hat{L}\{h(X_s)\} \dd{s} \bigg| X_0 \bigg]
+\end{aligned}$$
+
+For suffiently small $t$, the integral can be replaced by its first-order approximation:
+
+$$\begin{aligned}
+    \mathbf{E}[Y_t | X_0]
+    \approx Y_0 + \hat{L}\{h(X_0)\} \: t
+\end{aligned}$$
+
+Rearranging this gives the following,
+to be understood in the limit $t \to 0^+$:
+
+$$\begin{aligned}
+    \hat{L}\{h(X_0)\}
+    \approx \frac{1}{t} \mathbf{E}[Y_t - Y_0| X_0]
+\end{aligned}$$
+</div>
+</div>
+
+The general definition of resembles that of a classical derivative,
+and indeed, the generator $\hat{A}$ can be thought of as a differential operator.
+In that case, we would like an analogue of the classical
+fundamental theorem of calculus to relate it to integration.
+
+Such an analogue is provided by **Dynkin's formula**:
+for a stopping time $\tau$ with a finite expected value $\mathbf{E}[\tau|X_0] < \infty$,
+it states that:
+
+$$\begin{aligned}
+    \boxed{
+        \mathbf{E}\big[ h(X_\tau) | X_0 \big]
+        = h(X_0) + \mathbf{E}\bigg[ \int_0^\tau \hat{L}\{h(X_t)\} \dd{t} \bigg| X_0 \bigg]
+    }
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-dynkin"/>
+<label for="proof-dynkin">Proof</label>
+<div class="hidden">
+<label for="proof-dynkin">Proof.</label>
+The proof is similar to the one above.
+Define $Y_t = h(X_t)$ and use Itō’s lemma:
+
+$$\begin{aligned}
+    \dd{Y_t}
+    &= \bigg( \pdv{h}{x} f(X_t) + \frac{1}{2} \pdv[2]{h}{x} g^2(X_t) \bigg) \dd{t} + \pdv{h}{x} g(X_t) \dd{B_t}
+    \\
+    &= \hat{L} \{h(X_t)\} \dd{t} + \pdv{h}{x} g(X_t) \dd{B_t}
+\end{aligned}$$
+
+And then integrate this from $t = 0$ to the provided stopping time $t = \tau$:
+
+$$\begin{aligned}
+    Y_\tau
+    = Y_0 + \int_0^\tau \hat{L}\{h(X_t)\} \dd{t} + \int_0^\tau \pdv{h}{x} g(X_t) \dd{B_t}
+\end{aligned}$$
+
+All [Itō integrals](/know/concept/ito-integral/)
+are [martingales](/know/concept/martingale/),
+so the latter integral's conditional expectation is zero for the "initial" condition $X_0$.
+The rest of the above equality is also a martingale:
+
+$$\begin{aligned}
+    0
+    = \mathbf{E}\bigg[ Y_\tau - Y_0 - \int_0^\tau \hat{L}\{h(X_t)\} \dd{t} \bigg| X_0 \bigg]
+\end{aligned}$$
+
+Isolating this equation for $\mathbf{E}[Y_\tau | X_0]$ then gives Dynkin's formula.
+</div>
+</div>
+
+A common application of Dynkin's formula is predicting
+when the stopping time $\tau$ occurs, and in what state $X_\tau$ this happens.
+Consider an example:
+for a region $\Omega$ of state space with $X_0 \in \Omega$,
+we define the exit time $\tau \equiv \inf\{ t : X_t \notin \Omega \}$,
+provided that $\mathbf{E}[\tau | X_0] < \infty$.
+
+To get information about when and where $X_t$ exits $\Omega$,
+we define the *general reward* $\Gamma$ as follows,
+consisting of a *running reward* $R$ for $X_t$ inside $\Omega$,
+and a *terminal reward* $T$ on the boundary $\partial \Omega$ where we stop at $X_\tau$:
+
+$$\begin{aligned}
+    \Gamma
+    = \int_0^\tau R(X_t) \dd{t} + \: T(X_\tau)
+\end{aligned}$$
+
+For example, for $R = 1$ and $T = 0$, this becomes $\Gamma = \tau$,
+and if $R = 0$, then $T(X_\tau)$ can tell us the exit point.
+Let us now define $h(X_0) = \mathbf{E}[\Gamma | X_0]$,
+and apply Dynkin's formula:
+
+$$\begin{aligned}
+    \mathbf{E}\big[ h(X_\tau) | X_0 \big]
+    &= \mathbf{E}\big[ \Gamma \big| X_0 \big] + \mathbf{E}\bigg[ \int_0^\tau \hat{L}\{h(X_t)\} \dd{t} \bigg| X_0 \bigg]
+    \\
+    &= \mathbf{E}\big[ T(X_\tau) | X_0 \big] + \mathbf{E}\bigg[ \int_0^\tau \hat{L}\{h(X_t)\} + R(X_t) \dd{t} \bigg| X_0 \bigg]
+\end{aligned}$$
+
+The two leftmost terms depend on the exit point $X_\tau$,
+but not directly on $X_t$ for $t < \tau$,
+while the rightmost depends on the whole trajectory $X_t$.
+Therefore, the above formula is fulfilled
+if $h(x)$ satisfies the following equation and boundary conditions:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{cases}
+            \hat{L}\{h(x)\} + R(x) = 0 & \mathrm{for}\; x \in \Omega \\
+            h(x) = T(x) & \mathrm{for}\; x \notin \Omega
+        \end{cases}
+    }
+\end{aligned}$$
+
+In other words, we have just turned a difficult question about a stochastic trajectory $X_t$
+into a classical differential boundary value problem for $h(x)$.
+
+
+
+## References
+1.  U.H. Thygesen,
+    *Lecture notes on diffusions and stochastic differential equations*,
+    2021, Polyteknisk Kompendie.
diff --git a/content/know/concept/einstein-coefficients/index.pdc b/content/know/concept/einstein-coefficients/index.pdc
index 9feaf8c..f0f0f96 100644
--- a/content/know/concept/einstein-coefficients/index.pdc
+++ b/content/know/concept/einstein-coefficients/index.pdc
@@ -126,7 +126,7 @@ $$\begin{aligned}
 
 Note that this result holds even if $E_1$ is not the ground state,
 but instead some lower excited state below $E_2$,
-due to the principle of *detailed balance*.
+due to the principle of [detailed balance](/know/concept/detailed-balance/).
 Furthermore, it turns out that these relations
 also hold if the system is not in equilibrium.
 
diff --git a/content/know/concept/ito-calculus/index.pdc b/content/know/concept/ito-calculus/index.pdc
index 7a80e2f..3d4dd67 100644
--- a/content/know/concept/ito-calculus/index.pdc
+++ b/content/know/concept/ito-calculus/index.pdc
@@ -4,6 +4,7 @@ firstLetter: "I"
 publishDate: 2021-11-06
 categories:
 - Mathematics
+- Stochastic analysis
 
 date: 2021-11-06T14:34:00+01:00
 draft: false
diff --git a/content/know/concept/ito-integral/index.pdc b/content/know/concept/ito-integral/index.pdc
index cbd4a91..f923ca6 100644
--- a/content/know/concept/ito-integral/index.pdc
+++ b/content/know/concept/ito-integral/index.pdc
@@ -4,6 +4,7 @@ firstLetter: "I"
 publishDate: 2021-11-06
 categories:
 - Mathematics
+- Stochastic analysis
 
 date: 2021-10-21T19:41:58+02:00
 draft: false
diff --git a/content/know/concept/kolmogorov-equations/index.pdc b/content/know/concept/kolmogorov-equations/index.pdc
index 331d803..a3b11db 100644
--- a/content/know/concept/kolmogorov-equations/index.pdc
+++ b/content/know/concept/kolmogorov-equations/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-11-14
 categories:
 - Mathematics
 - Statistics
+- Stochastic analysis
 
 date: 2021-11-13T21:05:30+01:00
 draft: false
@@ -201,6 +202,48 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
+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
diff --git a/content/know/concept/markov-process/index.pdc b/content/know/concept/markov-process/index.pdc
index 536aa00..8aeb2b2 100644
--- a/content/know/concept/markov-process/index.pdc
+++ b/content/know/concept/markov-process/index.pdc
@@ -4,6 +4,7 @@ firstLetter: "M"
 publishDate: 2021-11-14
 categories:
 - Mathematics
+- Stochastic analysis
 
 date: 2021-11-13T21:05:21+01:00
 draft: false
diff --git a/content/know/concept/martingale/index.pdc b/content/know/concept/martingale/index.pdc
index 41c2709..ee39664 100644
--- a/content/know/concept/martingale/index.pdc
+++ b/content/know/concept/martingale/index.pdc
@@ -4,6 +4,7 @@ firstLetter: "M"
 publishDate: 2021-10-31
 categories:
 - Mathematics
+- Stochastic analysis
 
 date: 2021-10-18T10:01:46+02:00
 draft: false
diff --git a/content/know/concept/random-variable/index.pdc b/content/know/concept/random-variable/index.pdc
index bc41744..6ad4159 100644
--- a/content/know/concept/random-variable/index.pdc
+++ b/content/know/concept/random-variable/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-10-22
 categories:
 - Mathematics
 - Statistics
+- Measure theory
 
 date: 2021-10-21T20:40:42+02:00
 draft: false
diff --git a/content/know/concept/sigma-algebra/index.pdc b/content/know/concept/sigma-algebra/index.pdc
index 94e7306..6d7d38c 100644
--- a/content/know/concept/sigma-algebra/index.pdc
+++ b/content/know/concept/sigma-algebra/index.pdc
@@ -4,6 +4,7 @@ firstLetter: "S"
 publishDate: 2021-10-22
 categories:
 - Mathematics
+- Measure theory
 
 date: 2021-10-18T10:01:35+02:00
 draft: false
diff --git a/content/know/concept/stochastic-process/index.pdc b/content/know/concept/stochastic-process/index.pdc
index 5d50da8..1e500cc 100644
--- a/content/know/concept/stochastic-process/index.pdc
+++ b/content/know/concept/stochastic-process/index.pdc
@@ -4,6 +4,8 @@ firstLetter: "S"
 publishDate: 2021-11-07
 categories:
 - Mathematics
+- Stochastic analysis
+- Measure theory
 
 date: 2021-11-07T18:45:42+01:00
 draft: false
diff --git a/content/know/concept/wiener-process/index.pdc b/content/know/concept/wiener-process/index.pdc
index dc3892d..11c7a6e 100644
--- a/content/know/concept/wiener-process/index.pdc
+++ b/content/know/concept/wiener-process/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-10-29
 categories:
 - Physics
 - Mathematics
+- Stochastic analysis
 
 date: 2021-10-21T19:40:02+02:00
 draft: false