Abstract
Let (St)t≥0 be the running maximum of a standard Brownian motion (Bt)t≥0 and Tm≔inf{t;mSt<t},m>0. In this note we calculate the joint distribution of Tm and BTm. The motivation for our work comes from a mathematical model for animal foraging. We also present results for Brownian motion with drift.
Highlights
A part of the motivation behind the study presented here stems from a toymodel designed by Paul Krapivsky for animal foraging [4]
The animal's t initial position coincides with the origin, and we model its position as time elapses by a standard Brownian motion (Bt)t≥0
The third approach is to study the problem in a discrete setting and anticipate a passage to limit to obtain the formula for standard Brownian motion
Summary
A part of the motivation behind the study presented here stems from a toymodel designed by Paul Krapivsky for animal foraging [4]. The probability that the forager survives up to a time t is given by the probability that Ss ≥ s for all s ∈ [0, t] The proof for standard Brownian motion does not, explain how the distribution was originally found. The third approach is to study the problem in a discrete setting and anticipate a passage to limit to obtain the formula for standard Brownian motion. In all these approaches there are some technical diculties which we have not been able to resolve up to now.
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