# Approximation of stopped Brownian local time by diadic crossing chains

## Author

Listed:
• Knight, Frank B.

## Abstract

Let B(t) be a Brownian motion on R, B(0) = 0, and for [alpha]n:= 2-n let Tn0 = 0, Tnk+1 = inf{t> Tnk:B(t)-B(Tnk) = [alpha]n}, 0 [less-than-or-equals, slant] k. Then B(Tnk):= Rn(k[alpha]2n) 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 [sigma](Rn), and 2. (b) the estimator E(L(Â·)[sigma](Rn)). Let Nn(k) denote the number of upcrossings by Rn of (k[alpha]n, (k + 1)[alpha]n) by step Mn. Explicit formulae for (a) and (b) are obtained in terms of Nn. More generally, for T = TnKn, 0[less-than-or-equals, slant]Kn [set membership, variant] [sigma](Rn), let L(x) be the local time at T, and let NÂ±n(k) be the respective numbers of upcrossings (downcrossings) by step Kn. Simple expressions for (a) and (b) are given in terms of NÂ±n. For fixed measure [mu] on R, 2nE[[integral operator](E(L(x)[sigma](Rn)) - L(x))2[mu](dx)[sigma](Rn] is obtained, and when [mu](dx) = dx it reduces to . With T kept fixed as n --> [infinity], this converges P-a.s. to .

## Suggested Citation

• Knight, Frank B., 1997. "Approximation of stopped Brownian local time by diadic crossing chains," Stochastic Processes and their Applications, Elsevier, vol. 66(2), pages 253-270, March.
• Handle: RePEc:eee:spapps:v:66:y:1997:i:2:p:253-270
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File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(96)00119-6

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## References listed on IDEAS

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1. Mao, Xuerong, 1995. "Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 58(2), pages 281-292, August.
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## Citations

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Cited by:

1. Ohashi, Alberto & Simas, Alexandre B., 2015. "A note on the sharp Lp-convergence rate of upcrossings to the Brownian local time," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 137-141.
2. Leão, Dorival & Ohashi, Alberto, 2010. "Weak Approximations for Wiener Functionals," Insper Working Papers wpe_215, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.

### Keywords

Brownian motion Approximating random walks Local time Bessel processes Conditional mean squared error Upcrossings;

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