This is not financial advice.

Martingales are important to quantitative finance. A process $M_i$ is a martingale with respect to a measure $\mathbf P$ and a filtration $\mathcal F_i$ if

$$\langle M_j\rangle_{\mathbf P,\mathcal F_i}=M_i,\quad\forall i\leq j$$

Let's decode this definition. The $\left<\text{bra ket}\right>$ is an expectation operator. A filtration $\mathcal F_i$ is the history of a security up until the $i$th time tick. The fact that a martingale is a function of a filtration implies that a martingale is a stochastic process. Fix a measure $\mathbf P$. Our equation states that the expected value of a martingale is equal to its current value.

A process which is a martingale with respect to measure $\mathbf P$ is called a $\mathbf P$-martingale.

The Tower Law

The tower laws says that the present expected value of a future expected value is the present expected value.

$$\left\langle\langle X\rangle_{\mathbf P,\mathcal F_j}\right\rangle_{\mathbf P,\mathcal F_i}=\langle X\rangle_{\mathbf P,\mathcal F_j}$$

A claim is an abstraction of ownership that can incorporate arbitrary derivatives. A claim is a function of a filtration $\mathcal F_T$ over some time horizon $T$. The expectation value of a claim is a martingale.

For more information on this subject, check out Chapter 6 of the free ebook Phynance.