From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/martingale/index.md | 36 ++++++++++++++++-----------------
 1 file changed, 18 insertions(+), 18 deletions(-)

(limited to 'source/know/concept/martingale')

diff --git a/source/know/concept/martingale/index.md b/source/know/concept/martingale/index.md
index a54320f..9d3c6b4 100644
--- a/source/know/concept/martingale/index.md
+++ b/source/know/concept/martingale/index.md
@@ -13,25 +13,25 @@ A **martingale** is a type of
 with important and useful properties,
 especially for stochastic calculus.
 
-For a stochastic process $\{ M_t : t \ge 0 \}$
-on a probability filtered space $(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, P)$,
-then $M_t$ is a martingale if it satisfies all of the following:
-
-1.  $M_t$ is $\mathcal{F}_t$-adapted, meaning
-    the filtration $\mathcal{F}_t$ contains enough information
-    to reconstruct the current and all past values of $M_t$.
-2.  For all times $t \ge 0$, the expectation value exists $\mathbf{E}(M_t) < \infty$.
-3.  For all $s, t$ satisfying $0 \le s \le t$,
+For a stochastic process $$\{ M_t : t \ge 0 \}$$
+on a probability filtered space $$(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, P)$$,
+then $$M_t$$ is a martingale if it satisfies all of the following:
+
+1.  $$M_t$$ is $$\mathcal{F}_t$$-adapted, meaning
+    the filtration $$\mathcal{F}_t$$ contains enough information
+    to reconstruct the current and all past values of $$M_t$$.
+2.  For all times $$t \ge 0$$, the expectation value exists $$\mathbf{E}(M_t) < \infty$$.
+3.  For all $$s, t$$ satisfying $$0 \le s \le t$$,
     the [conditional expectation](/know/concept/conditional-expectation/)
-    $\mathbf{E}(M_t | \mathcal{F}_s) = M_s$,
-    meaning the increment $M_t \!-\! M_s$ is always expected
-    to be zero $\mathbf{E}(M_t \!-\! M_s | \mathcal{F}_s) = 0$.
+    $$\mathbf{E}(M_t | \mathcal{F}_s) = M_s$$,
+    meaning the increment $$M_t \!-\! M_s$$ is always expected
+    to be zero $$\mathbf{E}(M_t \!-\! M_s | \mathcal{F}_s) = 0$$.
 
 The last condition is called the **martingale property**,
 and basically means that a martingale is an unbiased random walk.
-Accordingly, the [Wiener process](/know/concept/wiener-process/) $B_t$
+Accordingly, the [Wiener process](/know/concept/wiener-process/) $$B_t$$
 (Brownian motion) is an example of a martingale,
-since each of its increments $B_t \!-\! B_s$ has mean $0$ by definition.
+since each of its increments $$B_t \!-\! B_s$$ has mean $$0$$ by definition.
 
 Martingales are easily confused with
 [Markov processes](/know/concept/markov-process/),
@@ -42,15 +42,15 @@ the martingale property says nothing about history-dependence,
 and the Markov property does not say *what* the future expectation should be.
 
 Modifying property (3) leads to two common generalizations.
-The stochastic process $M_t$ above is a **submartingale**
+The stochastic process $$M_t$$ above is a **submartingale**
 if the current value is a lower bound for the expectation:
 
-3.  For $0 \le s \le t$, the conditional expectation $\mathbf{E}(M_t | \mathcal{F}_s) \ge M_s$.
+3.  For $$0 \le s \le t$$, the conditional expectation $$\mathbf{E}(M_t | \mathcal{F}_s) \ge M_s$$.
 
-Analogouly, $M_t$ is a **supermartingale**
+Analogouly, $$M_t$$ is a **supermartingale**
 if the current value is an upper bound instead:
 
-3.  For $0 \le s \le t$, the conditional expectation $\mathbf{E}(M_t | \mathcal{F}_s) \le M_s$.
+3.  For $$0 \le s \le t$$, the conditional expectation $$\mathbf{E}(M_t | \mathcal{F}_s) \le M_s$$.
 
 Clearly, submartingales and supermartingales are *biased* random walks,
 since they will tend to increase and decrease with time, respectively.
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