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Diffstat (limited to 'source/know/concept/martingale/index.md')
| -rw-r--r-- | source/know/concept/martingale/index.md | 6 |
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. |