Local martingale

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In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property.

[edit] Definition

Let (Ω, F, P) be a probability space; let F_∗ = { F_t | t ≥ 0 } be a filtration of F; let X : [0, +∞) × Ω → S be an F_∗-adapted stochastic process taking values in a measurable space (S, Σ). Then X is called an F_∗-local martingale if there exists a sequence of F_∗-stopping times τ_k : Ω → [0, +∞) such that

• the τ_k are almost surely increasing: P[τ_k < τ_k+1] = 1;
• the τ_k diverge almost surely: P[τ_k → +∞ as k → +∞] = 1;
• the stopped process

is an F_∗-martingale for every k.

[edit] References

• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications, Sixth edition, Berlin: Springer. ISBN 3-540-04758-1.