Categories: Mathematics, Stochastic analysis.


A martingale is a type of stochastic process with important and useful properties, especially for stochastic calculus.

For a stochastic process {Mt:t0}\{ M_t : t \ge 0 \} on a probability filtered space (Ω,F,{Ft},P)(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, P), then MtM_t is a martingale if it satisfies all of the following:

  1. MtM_t is Ft\mathcal{F}_t-adapted, meaning the filtration Ft\mathcal{F}_t contains enough information to reconstruct the current and all past values of MtM_t.
  2. For all times t0t \ge 0, the expectation value exists E(Mt)<\mathbf{E}(M_t) < \infty.
  3. For all s,ts, t satisfying 0st0 \le s \le t, the conditional expectation E(MtFs)=Ms\mathbf{E}(M_t | \mathcal{F}_s) = M_s, meaning the increment Mt ⁣ ⁣MsM_t \!-\! M_s is always expected to be zero E(Mt ⁣ ⁣MsFs)=0\mathbf{E}(M_t \!-\! M_s | \mathcal{F}_s) = 0.

The last condition is called the martingale property, and basically means that a martingale is an unbiased random walk. Accordingly, the Wiener process BtB_t (Brownian motion) is an example of a martingale, since each of its increments Bt ⁣ ⁣BsB_t \!-\! B_s has mean 00 by definition.

Martingales are easily confused with Markov processes, because stochastic processes will often be both, e.g. the Wiener process. However, these are distinct concepts: the martingale property says nothing about history-dependence, and the Markov property does not say what the future expectation should be.

Modifying property (3) leads to two common generalizations. The stochastic process MtM_t above is a submartingale if the current value is a lower bound for the expectation:

  1. For 0st0 \le s \le t, the conditional expectation E(MtFs)Ms\mathbf{E}(M_t | \mathcal{F}_s) \ge M_s.

Analogouly, MtM_t is a supermartingale if the current value is an upper bound instead:

  1. For 0st0 \le s \le t, the conditional expectation E(MtFs)Ms\mathbf{E}(M_t | \mathcal{F}_s) \le M_s.

Clearly, submartingales and supermartingales are biased random walks, since they will tend to increase and decrease with time, respectively.


  1. U.H. Thygesen, Lecture notes on diffusions and stochastic differential equations, 2021, Polyteknisk Kompendie.