# Approximation of stopped Brownian local time by diadic crossing chains

## Abstract

AbstractLet B(t) be a Brownian motion on R, B(0) = 0, and for αn:= 2−n let Tn0 = 0, Tnk+1 = inf{t > Tnk:|B(t)−B(Tnk)| = αn}, 0 ⩽ k. Then B(Tnk):= Rn(kα2n) is the nth approximating random walk. Define Mn by TnMn = T(−1) (the passage time to −1) and let L(x) be the local time of B at T(−1). The paper is concerned with 1.(a) the conditional law of L given σ(Rn), and2.(b) the estimator E(L(·)|σ(Rn)).Let Nn(k) denote the number of upcrossings by Rn of (kαn, (k + 1)αn) by step Mn. Explicit formulae for (a) and (b) are obtained in terms of Nn.More generally, for T = TnKn, 0⩽Kn ∈ σ(Rn), let L(x) be the local time at T, and let N±n(k) be the respective numbers of upcrossings (downcrossings) by step Kn. Simple expressions for (a) and (b) are given in terms of N±n. For fixed measure μ on R, 2nE[∫(E(L(x)|σ(Rn)) − L(x))2μ(dx)|σ(Rn] is obtained, and when μ(dx) = dx it reduces to 1415α2nKn. With T kept fixed as n → ∞, this converges P-a.s. to 1415T