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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,{ℱ_t},ℱ,P) should fulfil the following:

1. X(t) is adapted to ℱ_t for all t≥0.
2. E[|X(t)|]<∞ for all t≥0.
3. E[X(T)|ℱ_t]=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
  Y_λ(t)=e^{λX(t)-λ₀∫t μ(X(s))ds-λ2/2₀∫t σ(X(s))2ds},
  is a martingale for all λ∈ℝ such that
  E[e^{(λ2/2₎₀∫t σ(X(s))2ds}]<∞. If the last condition is not fulfilled then Y_λ(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
  e^{λW(t)-λ2t/2}
  is martingale for all λ∈ℂ.