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authorPrefetch2022-11-08 18:14:21 +0100
committerPrefetch2022-11-08 18:14:21 +0100
commit5ed7553b723a9724f55e75261efe2666e75df725 (patch)
tree2d893dbe47b11a569a4de12dba05b9eac35f6350 /source/know/concept/martingale
parent70006b2c540543a96e54254823f95348e9f0ed7a (diff)
The tweaks and fixes never stop
Diffstat (limited to 'source/know/concept/martingale')
-rw-r--r--source/know/concept/martingale/index.md6
1 files changed, 4 insertions, 2 deletions
diff --git a/source/know/concept/martingale/index.md b/source/know/concept/martingale/index.md
index 9d3c6b4..53a346a 100644
--- a/source/know/concept/martingale/index.md
+++ b/source/know/concept/martingale/index.md
@@ -45,12 +45,14 @@ Modifying property (3) leads to two common generalizations.
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**
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.