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On oscillations of the geometric Brownian motion with time-delayed drift

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  • Gushchin, Alexander A.
  • Küchler, Uwe

Abstract

The geometric Brownian motion is the solution of a linear stochastic differential equation in the Itô sense. If one adds to the drift term a possible nonlinear time-delayed term and starts with a non-negative initial process then the process generated in this way, may hit zero and may oscillate around zero infinitely many times depending on properties of both the drift terms and the diffusion constant.

Suggested Citation

  • Gushchin, Alexander A. & Küchler, Uwe, 2004. "On oscillations of the geometric Brownian motion with time-delayed drift," Statistics & Probability Letters, Elsevier, vol. 70(1), pages 19-24, October.
    • Handle: RePEc:eee:stapro:v:70:y:2004:i:1:p:19-24
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    References listed on IDEAS

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    1. Appleby, John A. D. & Buckwar, Evelyn, 2003. "Noise Induced Oscillation in Solutions of Stochastic Delay Differential Equations," SFB 373 Discussion Papers 2003,9, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    2. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48, January.
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