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Conditioning the tanh-drift process on first-passage times: Exact drifts, bridges, and process equivalences

Kacim François-Élie, Alain Mazzolo

Abstract

In this article, we consider the Beneš process with drift $μ(x)=α\tanh(αx + β)$, with $α> 0$, $β\in \mathbb{R}$, and an absorbing barrier at $x=a$. After deriving the propagator and key associated quantities -- the first-passage-time distribution and the survival probability -- we then condition this process to have various prescribed first-passage-time distributions. When the conditioning is imposed at an infinite time horizon, this procedure reveals the existence of different processes that share the same first-passage-time distribution as the Beneš process, a phenomenon recently observed in the case of Brownian motion with drift. When the conditioning is imposed at a finite time horizon, the procedure shows that the conditioned Beneš process and the Brownian motion with drift under the same conditioning exhibit identical behaviors. This strengthens an elegant result of Benjamini and Lee stating that Brownian motion and the Beneš process share the same Brownian bridge, and it also connects with more recent findings obtained by conditioning two independent identical Brownian motions with drift, or two independent Beneš processes that annihilate upon meeting. Moreover, we show that several conditioned Beneš drifts converge near the absorbing boundary to the drift of the taboo diffusion, which motivates a parallel analysis of the taboo process itself. Using Girsanov's theorem, we derive its propagator, first-passage-time distribution, and conditioned versions, thereby further clarifying the structural relationships between Beneš, Brownian, and taboo dynamics.

Conditioning the tanh-drift process on first-passage times: Exact drifts, bridges, and process equivalences

Abstract

In this article, we consider the Beneš process with drift , with , , and an absorbing barrier at . After deriving the propagator and key associated quantities -- the first-passage-time distribution and the survival probability -- we then condition this process to have various prescribed first-passage-time distributions. When the conditioning is imposed at an infinite time horizon, this procedure reveals the existence of different processes that share the same first-passage-time distribution as the Beneš process, a phenomenon recently observed in the case of Brownian motion with drift. When the conditioning is imposed at a finite time horizon, the procedure shows that the conditioned Beneš process and the Brownian motion with drift under the same conditioning exhibit identical behaviors. This strengthens an elegant result of Benjamini and Lee stating that Brownian motion and the Beneš process share the same Brownian bridge, and it also connects with more recent findings obtained by conditioning two independent identical Brownian motions with drift, or two independent Beneš processes that annihilate upon meeting. Moreover, we show that several conditioned Beneš drifts converge near the absorbing boundary to the drift of the taboo diffusion, which motivates a parallel analysis of the taboo process itself. Using Girsanov's theorem, we derive its propagator, first-passage-time distribution, and conditioned versions, thereby further clarifying the structural relationships between Beneš, Brownian, and taboo dynamics.

Paper Structure

This paper contains 27 sections, 108 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Girsanov Theorem
  3. Motivations for using Girsanov’s theorem
  4. Girsanov’s theorem in practice
  5. First example: from a standard Brownian motion to a Brownian motion with constant drift
  6. Second example: from a Brownian motion with an absorbing barrier at $a$ to a Brownian motion with drift and the same absorbing barrier.
  7. The tanh-drift process
  8. Conditioning the tanh-drift process
  9. Conditioning on a finite time horizon $T$
  10. Conditioning on full survival as $t \to \infty$
  11. Conditioning on a prescribed first-passage time distribution
  12. Conditioning with respect to the first-passage time density $\gamma^*(t) = \frac{a }{\sqrt{2 \pi t^3} } e^{- \frac{(a-\mu t )^2}{2 t}}$
  13. Conditioning with respect to the first-passage time density $\gamma^*(t) = \frac{a }{\sqrt{2 \pi t^3} } \frac{\cosh(\gamma a + \delta)}{\cosh(\delta)} e^{-\frac{a^2+\gamma ^2 t^2}{2 t}}$
  14. Conditioning the drifted Brownian motion
  15. Conditioning the Brownian motion with drift $\mu$ with respect to the first-passage time density of the tanh-drift process $\gamma^*(t) = \frac{a }{\sqrt{2 \pi t^3} } \frac{\cosh(\alpha a + \beta)}{\cosh(\beta)} e^{-\frac{a^2+\alpha ^2 t^2}{2 t}}$
  16. ...and 12 more sections

Figures (1)

  • Figure 1: Plot of the conditioned drift with the parameter $\mu = -1$ and of the original tanh-drift process with the parameters $\alpha = 1$ and $\beta = 0$, both with the barrier located at $a = 5$. The two drifts exhibit completely different behaviors for positive positions, which in practice could make it possible to distinguish the two processes.