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On oscillations of the geometric Brownian motion with time delayed drift

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  • Küchler, Uwe
  • Gushchin, Alexander A.

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 nonnegative initial process then the process generated in this way, may hit zero and may oscillate around zero infinitely often depending on properties of both drift terms and the diffusion constant.

Suggested Citation

  • Küchler, Uwe & Gushchin, Alexander A., 2003. "On oscillations of the geometric Brownian motion with time delayed drift," SFB 373 Discussion Papers 2003,8, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    • Handle: RePEc:zbw:sfb373:20038
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    References listed on IDEAS

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    1. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48, January.
    2. 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.
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