Definition. Let $(\Omega, \F,(\F_t)_{t\in\mathbb{T}},\Prob)$ be a filtered probability space and $(X_t)$ be a stochastic process such that $X_t$ is integrable for all $t\in\mathbb{T}$ . Then, $X=(X_t, \F_t)$ is called a submartingale if $$\mathbb{E}^{\Prob}[X_t|\F_s] \geq X_s,\, \mbox{for every $s < t$, a.e.[$\Prob$],}$$ and a supermartigale if $$\mathbb{E}^{\Prob}[X_t|\F_s] \leq X_s,\, \mbox{for every $s < t$, a.e.[$\Prob$].}$$

A submartingale that is also a supermartingale is called a martingale, i.e., a martingale satisfies $$\mathbb{E}^{\Prob}[X_t|\F_s] = X_s,\, \mbox{for every $s < t$, a.e.[$\Prob$].}$$

Similarly, if the $\{\F_t\}$ form a decreasing collection of $\sigma$ -subalgebras of $\F$ , then $X$ is called a reverse submartingale if $$\mathbb{E}^{\Prob}[X_s|\F_t] \geq X_t,\, \mbox{for every $s < t$, a.e.[$\Prob$],}$$ and a reverse supermartingale if $$\mathbb{E}^{\Prob}[X_s|\F_t] \leq X_t,\, \mbox{for every $s < t$, a.e.[$\Prob$].}$$

Remarks

• The martingale property captures the idea of a fair bet, where the expected future value is equal to the current value.
• The submartingale property is equivalent to $$\int_A X_t \, d\Prob \geq \int_A X_s \, d\Prob \,\,\, \mbox{for every $A \in \F_s$ and $s < t$}$$ and similarly for the other definitions. This is immediate from the definition of conditional expectation.