Categories: Mathematics, Stochastic analysis.

# Martingale

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

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

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

The last condition is called the martingale property, and basically means that a martingale is an unbiased random walk. Accordingly, the Wiener process $$B_t$$ (Brownian motion) is an example of a martingale, since each of its increments $$B_t \!-\! B_s$$ has mean $$0$$ 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 $$M_t$$ above is a submartingale if the current value is a lower bound for the expectation:

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

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

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

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

## References

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

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.