> [!theorem] > > Let $\seq{X_n}$ be a discrete time $\seq{\cf_n}$-[[Martingale|submartingale]], then there exists $\seq{\cf_n}$-adapted processes $\seq{A_n}$ and $\seq{M_n}$ such that > 1. $X_n = A_n + M_n$. > 2. $A_0 = 0$, $A_n \le A_{n+1}$, and $A_n$ is $\cf_{n-1}$-measurable for all $n \in \nat$. > 3. $M_n$ is a martingale. > > Moreover, the decomposition is unique almost surely. > > *Proof*. For any $n \in \nat$, let > $ > A_n = \sum_{j = 1}^n\ev\braks{X_j|\cf_{j-1}} - X_{j-1} > $ > then $A_n$ satisfies the desired properties. Define $M_n = X_n - A_n$, then > $ > \begin{align*} > \ev\braks{M_n|\cf_{n-1}} &= \ev\braks{X_n|\cf_{n - 1}} - A_n \\ > &= X_{n-1} - A_{n-1} = M_{n-1} > \end{align*} > $ > [!theorem] > > Let $\bracs{X_t: t \ge 0}$ be a continuous time, non-negative $\bracs{\cf_t}$-submartingale with [[RCLL]]/continuous sample paths. Then there exists two continuous time [[Stochastic Process|stochastic processes]] such that: > 1. $X_t = A_t + M_t$ for all $t \ge 0$. > 2. $A_0 = 0$, $A_t$ has nondecreasing, RCLL/continuous sample paths. > 3. $A_t$ is $\bracs{\cf_{t^-}}$-measurable. > 4. $M_t$ is a $\bracs{\cf_t}$-martingale with RCLL/continuous sample paths. > > Moreover, this decomposition is unique up to modifications. If $X$ is uniformly integrable, then $A$ and $M$ are both uniformly integrable.