Penalisation of the Standard Random Walk by a Function of the One-Sided Maximum, of the Local Time, or of the Duration of the Excursions

Abstract

Call (Ω,F _∞,ℙ,X,F) the canonical space for the standard random walk on ℤ. Thus, Ω denotes the set of paths φ : ℕ → ℤ such that ∣φ(n + 1) − φ(n)∣ = 1, X = (X _n , n ≥ 0) is the canonical coordinate process on Ω; F = (F _n , n ≥ 0) is the natural filtration of X, F _∞ the σ-field V _n ≥0 F _n , and ℙ₀ the probabilitiy on (Ω,F _∞) such that under ℙ₀, X is the standard random walk started form 0, i.e., ℙ₀ (X _n+1 = j ∣X _n = i) = ½ when ∣j − i∣ = 1.

Let G : ℕ × Ω → ℝ⁺ be a positive, adapted functional. For several types of functionals G, we show the existence of a positive F-martingale (M _n , n ≥ 0) such that, for all n and all Λ _n ∈ F _n ,

$$\frac{{{\mathbb E}_{\rm 0} [1_{\Lambda _n } G_p ]}}{{{\mathbb E}_{\rm 0} [G_p ]}} \to {\mathbb E}_{\rm 0} [1_{\Lambda _n } M_n ] \quad {\rm when} \quad p \to \infty.$$

Thus, there exists a probability Q on (Ω,F _∞) such that Q(Λ _n ) = ${\mathbb E}_{\rm 0} [1_{\Lambda _n } M_n ]$ for all Λ _n ∈ F _n . We describe the behavior of the process (Ω,X,F) under Q.

The three sections of the article deal respectively with the three situations when G is a function:

• of the one-sided maximum;

• of the sign of X and of the time spent at zero;

• of the length of the excursions of X.