Martingales

A martingale is a special kind of stochastic process that captures the idea of a fair game.

Definition

To be added...

Intuition

• Martingales represent pure randomness with no drift.
• They generalize the role of constant functions in the deterministic setting:
  • In ODEs, “no derivative” means constant.
  • In SDEs, “no drift term” means martingale.

Stochastic derivative (Itô sense)

In stochastic calculus, processes can be decomposed as

• The coefficient ${\mu }_t$ is the drift (deterministic trend).
• The term ${\sigma }_{t}d{W}_t$ is the martingale part (unpredictable noise).

A martingale is exactly a process with stochastic derivative equal to pure noise:

so its evolution has no predictable drift.

Examples

• Constant process: $M_t\equiv c$.
• Brownian motion $B_t$.
• Compensated Poisson process: $N_t-\lambda t$.

Key idea

Martingales are the constants of stochastic calculus:

• Deterministic constant → no change at all.
• Martingale → no predictable change, only random fluctuations.