From 5ed7553b723a9724f55e75261efe2666e75df725 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Tue, 8 Nov 2022 18:14:21 +0100 Subject: The tweaks and fixes never stop --- source/know/concept/martingale/index.md | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) (limited to 'source/know/concept/martingale') 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. -- cgit v1.2.3