Approximation of stopped Brownian local time by diadic crossing chains

[摘要] Let B(t) be a Brownian motion on R, B(0) = 0, and for alpha(n):= 2(-n) let T-0(n) = 0, T-k+1(n) = inf{t > T-k(n): \B(t)-B(T-k(n))\ = alpha(n)}, 0 less than or equal to k. Then B(T-k(n)) := R(n)(k alpha(n)(2)) is the nth approximating random walk. Define M(n) by T-Mn(n) = 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 (a) the conditional law of L given sigma(R(n)), and (b) the estimator E(L(.)\sigma(R(n))). Let N-n(k) denote the number of upcrossings by R(n) of (k alpha(n), (k + 1)alpha(n)) by step M(n). Explicit formulae for (a) and (b) are obtained in terms of N-n. More generally, for T = T-Kn(n), 0 less than or equal to K-n is an element of sigma(R(n)), let L(x) be the local time at T, and let N-n(+/-)(k) be the respective numbers of upcrossings (downcrossings) by step K-n. Simple expressions for (a) and (b) are given in terms of N-n(+/-). For fixed measure mu on R, 2(n)E[integral(E(L(x)\alpha(R(n))) - L(x))(2) mu(dx)\sigma(R(n))] is obtained, and when mu(dx) = dx it reduces to 14/15 alpha(n)(2)K(n). With T kept fixed as n --> infinity, this converges P-a.s. to 14/15T.

[发布日期] 1997-03-28

[关键词] Brownian motion;approximating random walks;local time;Bessel processes;conditional mean squared error;upcrossings