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A diffusion approximation result for two parameter processes
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  • Published: September 1994

A diffusion approximation result for two parameter processes

  • René A. Carmona1 &
  • Jean Pierre Fouque2 

Probability Theory and Related Fields volume 98, pages 277–298 (1994)Cite this article

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Summary

We consider a one-dimensional linear wave equation with a small mean zero dissipative field and with the boundary condition imposed by the so-called Goursat problem. In order to observe the effect of the randomness on the solution we perform a space-time rescaling and we rewrite the problem in a diffusion approximation form for two parameter processes. We prove that the solution converges in distribution toward the solution of a two-parameter stochastic differential equation which we identify. The diffusion approximation results for oneparameter processes are well known and well understood. In fact, the solution of the one-parameter analog of the problem we consider here is immediate. Unfortunately, the situation is much more complicated for two-parameter processes and we believe that our result is the first one of its kind.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of California at Irvine, 92717, Irvine, CA, USA

    René A. Carmona

  2. Ecole Polytechnique, CNRS-CMAP, F-91128, Palaiseau cedex, France

    Jean Pierre Fouque

Authors
  1. René A. Carmona
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  2. Jean Pierre Fouque
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Additional information

Partially supported by ONR N00014-91-J-1010

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Cite this article

Carmona, R.A., Fouque, J.P. A diffusion approximation result for two parameter processes. Probab. Th. Rel. Fields 98, 277–298 (1994). https://doi.org/10.1007/BF01192255

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  • Received: 15 May 1992

  • Revised: 28 July 1993

  • Issue Date: September 1994

  • DOI: https://doi.org/10.1007/BF01192255

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Mathematics Subject Classifications (1991)

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