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author | Prefetch | 2022-10-20 18:25:31 +0200 |
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committer | Prefetch | 2022-10-20 18:25:31 +0200 |
commit | 16555851b6514a736c5c9d8e73de7da7fc9b6288 (patch) | |
tree | 76b8bfd30f8941d0d85365990bcdbc5d0643cabc /source/know/concept/martingale | |
parent | e5b9bce79b68a68ddd2e51daa16d2fea73b84fdb (diff) |
Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'
Diffstat (limited to 'source/know/concept/martingale')
-rw-r--r-- | source/know/concept/martingale/index.md | 36 |
1 files changed, 18 insertions, 18 deletions
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. |