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Range of Brownian Motion with Drift

Author

Listed:
  • Etienne Tanré

    (INRIA, Projet OMEGA)

  • Pierre Vallois

    (Institut Élie Cartan)

Abstract

Let (B δ (t)) t ≥ 0 be a Brownian motion starting at 0 with drift δ > 0. Define by induction S 1=− inf t ≥ 0 B δ (t), ρ1 the last time such that B δ (ρ1)=−S 1, S 2=sup0≤ t ≤ρ 1 B δ (t), ρ2 the last time such that B δ (ρ2)=S 2 and so on. Setting A k =S k +S k+1; k ≥ 1, we compute the law of (A 1,...,A k ) and the distribution of (B δ (t+ρ l) − B δ (ρ l ); 0 ≤ t ≤ ρ l-1 − ρ l )2 ≤ l ≤ k for any k ≥ 2, conditionally on (A 1,...,A k ). We determine the law of the range R δ (t) of (B δ (s)) s≥ 0 at time t, and the first range time θδ (a) (i.e. θδ (a)=inf{t > 0; R δ (t) > a}). We also investigate the asymptotic behaviour of θ δ (a) (resp. R δ (t)) as a → ∞ (resp. t → ∞).

Suggested Citation

  • Etienne Tanré & Pierre Vallois, 2006. "Range of Brownian Motion with Drift," Journal of Theoretical Probability, Springer, vol. 19(1), pages 45-69, January.
    • Handle: RePEc:spr:jotpro:v:19:y:2006:i:1:d:10.1007_s10959-006-0012-7
    DOI: 10.1007/s10959-006-0012-7
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    References listed on IDEAS

    as
    1. Pierre Vallois & Charles Tapiero, 1997. "Range reliability in random walks," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 45(3), pages 325-345, October.
    2. Vallois, P., 1995. "Decomposing the Brownian path via the range process," Stochastic Processes and their Applications, Elsevier, vol. 55(2), pages 211-226, February.
    3. Glynn, Peter W., 1985. "On the range of a regenerative sequence," Stochastic Processes and their Applications, Elsevier, vol. 20(1), pages 105-113, July.
    4. Imhof, J. P., 1992. "A construction of the brownian path from BES3 pieces," Stochastic Processes and their Applications, Elsevier, vol. 43(2), pages 345-353, December.
    5. Paavo Salminen & Pierre Vallois, 2005. "On First Range Times of Linear Diffusions," Journal of Theoretical Probability, Springer, vol. 18(3), pages 567-593, July.
    Full references (including those not matched with items on IDEAS)

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