On the First Exit Time of Geometric Brownian Motion from Stochastic Exponential Boundaries

  • Tristan GuillaumeEmail author
Original Paper


This article deals with the boundary crossing probability of a geometric Brownian motion (GBM) process when the boundary itself is a GBM process. An exact formula is obtained for the probability that the first exit time of \( S\left( t \right) \) from the stochastic interval \( \left[ {H_{1} \left( t \right),H_{2} \left( t \right)} \right] \) is greater than a finite time \( T \) using a partial differential equation approach. Applications and numerical results are provided. The possibility of an extension to higher dimension is also discussed. In particular, the steps to obtain the probability that \( S_{1} \left( t \right) \), \( S_{2} \left( t \right) \) and \( S_{3} \left( t \right) \) remain above \( S_{4} \left( t \right) \), \( \forall 0 \le t \le T \), are outlined, while pointing out that the entailed numerical issues make the relevance of an analytical approach questionable.


Brownian motion Boundary crossing probability First passage time probability Stochastic boundary Linear parabolic equation Heat equation 



The author thanks the anonymous referees for their comments and suggestions which contributed to improve the initial manuscript.


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© Springer Nature India Private Limited 2018

Authors and Affiliations

  1. 1.Laboratoire ThemaUniversité de Cergy-PontoiseCergy-Pontoise CedexFrance

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