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

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Abstract

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.

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Acknowledgements

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

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  1. Laboratoire Thema, Université de Cergy-Pontoise, 33 boulevard du port, 95011, Cergy-Pontoise Cedex, France

    Tristan Guillaume

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  1. Tristan Guillaume
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Guillaume, T. On the First Exit Time of Geometric Brownian Motion from Stochastic Exponential Boundaries. Int. J. Appl. Comput. Math 4, 120 (2018). https://doi.org/10.1007/s40819-018-0556-0

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