diff options
Diffstat (limited to 'content/know/concept/martingale/index.pdc')
| -rw-r--r-- | content/know/concept/martingale/index.pdc | 20 |
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 $\mathbf{E}(M_t | \mathcal{F}_s) \le M_s$. |