Expand knowledge base, reorganize measure theory, update gitignore

1 files changed, 9 insertions, 11 deletions

diff --git a/content/know/concept/martingale/index.pdc b/content/know/concept/martingale/index.pdc
index 07ed1a4..21fa918 100644
--- a/content/know/concept/martingale/index.pdc
+++ b/content/know/concept/martingale/index.pdc
@@ -12,15 +12,14 @@ markup: pandoc
 
 # Martingale
 
-A **martingale** is a type of stochastic process
-(i.e. a time-indexed [random variable](/know/concept/random-variable/))
+A **martingale** is a type of
+[stochastic process](/know/concept/stochastic-process/)
 with important and useful properties,
 especially for stochastic calculus.
 
 For a stochastic process $\{ M_t : t \ge 0 \}$
-on a probability space $(\Omega, \mathcal{F}, P)$ with filtration $\{ \mathcal{F}_t \}$
-(see [$\sigma$-algebra](/know/concept/sigma-algebra/)),
-then $\{ M_t \}$ is a martingale if it satisfies all of the following:
+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
@@ -33,19 +32,18 @@ then $\{ M_t \}$ is a martingale if it satisfies all of the following:
     to be zero $\mathbf{E}(M_t \!-\! M_s | \mathcal{F}_s) = 0$.
 
 The last condition is called the **martingale property**,
-and essentially means that a martingale is an unbiased random walk.
-Accordingly, the [Wiener process](/know/concept/wiener-process/) $\{ B_t \}$
-(Brownian motion) is a prime example of a martingale
-(with respect to its own filtration),
+and basically means that a martingale is an unbiased random walk.
+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.
 
 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 $\mathbf{E}(M_t | \mathcal{F}_s) \ge M_s$.
 
-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 $\mathbf{E}(M_t | \mathcal{F}_s) \le M_s$.