A diffusion approximation result for two parameter processes |
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Authors: | René A Carmona Jean Pierre Fouque |
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Institution: | (1) Department of Mathematics, University of California at Irvine, 92717 Irvine, CA, USA;(2) Ecole Polytechnique, CNRS-CMAP, F-91128 Palaiseau cedex, France |
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Abstract: | 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.Partially supported by ONR N00014-91-J-1010 |
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Keywords: | 60H15 |
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