## Glossary

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

# Martingale

The key property of a process, X say, which is a martingale is that the best prediction of the value, X(T), at any future timepoint T given all information available now (time t) is X(t).

From a mathematical point of view a martingale defined on a filtered probability space (Ω,{ℱt},ℱ,P) should fulfil the following:
1. X(t) is adapted to ℱt for all t≥0.
2. E[|X(t)|]<∞ for all t≥0.
3. E[X(T)|ℱt]=X(t) for all T≥t for all t≥0.
It is the third condition that is the crucial property. The first two are to some extent techinical conditions.

## Examples

• The Brownian motion is the most important matingale in continuous time.
• Ito integrals where the integrand has finite second moment is another important class of martingales.
• Let X(t) be a solution to the stochastic differential equation dX(t)=μ(X(t))dt+σ(X(t))dW(t) . Now
Yλ(t)=eλX(t)-λ0t μ(X(s))ds-λ2/20t σ(X(s))2ds,
is a martingale for all λ∈ℝ such that
E[e2/2)0t σ(X(s))2ds]<∞. If the last condition is not fulfilled then Yλ(t) is only a local martingale.
• If let X(t)=W(t) i.e. we let μ≡0 and σ≡1 in the last example we can even obtain that
eλW(t)-λ2t/2
is martingale for all λ∈ℂ.